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0 (the sum starts with {\displaystyle |n|} 0000033547 00000 n For me, poles and zeros are important to transfer functions, which describe the relationship between inputs and outputs, and they can tell you about stabilizability and stability. 0000001828 00000 n Find approximate values for the gain at ω=10, 103, and 106 rad/s. pzmap is useful to visually identify such near-canceling pole-zero pairs to perform pole-zero simplification. 0000042877 00000 n {\displaystyle z\mapsto f(z)} Finite zeros are shown by a "o" on the diagram above. . 0000033099 00000 n If f is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros. 0000032575 00000 n 0000020744 00000 n (this is a consequence of the analytic property). This example uses: SerDes Toolbox; RF Toolbox; Open Live Script. This duality is fundamental for the study of meromorphic functions. Finite zeros are shown by a "o" on the diagram above. Note that the number of zeros and poles is always equal if you include poles and zeros at infinity. First, express the numerator as a polynomial, then find its roots to get the zeros: EDU>>numerator = [1 0 0] numerator = 1 0 0 EDU>>roots(numerator) ans = 0 0 So the zeros are both at $$z=0$$. So the pole-zero representation consists of: a constant term, k=3, zeros at s=-1 and s=-2, and; Matt L. Matt L. 69.6k 4 4 gold badges 54 54 silver badges 128 128 bronze badges $\endgroup$ $\begingroup$ If I multiplied the numerator and denominator by z, then wouldn't I end up with … The plot below shows the poles (marked as "x") and the zeros (marked as "o") of … {\displaystyle \phi (z).} Ignore the frequency response of M 1. n {\displaystyle \phi } {\displaystyle z_{0}} z For a general discussion of zeros and poles of such functions, see Pole–zero plot § Continuous-time systems. Solution The zeros are as follows: one at s = 0 and one at s =∝. ( The pole-zero representation consists of: a constant term, k=3, zeros at s=-1 and s=-2, and; poles at s=-1+j, s=-1-j and s=-3. 0000018681 00000 n … | = 0000029712 00000 n 0000042855 00000 n − By convention, the poles of the system are indicated in the plot by an X while the zeros are indicated by a circle or O. . The effect of zeros are not covered in detail in this module; however, it is important to note that the step response of a system with a pole is a combination of a step and an impulse . ��k*��f��;͸�x��T9���1�yTr"@/lc���~M�n�B����T��|N 0 meromorphic) in a neighbourhood of a point z of M if there is a chart Pole-Zero Analysis This chapter discusses pole-zero analysis of digital filters.Every digital filter can be specified by its poles and zeros (together with a gain factor). , there is no principal part), one has a zero of order | Here, there poles and zeros of CL1 are blue, and those of CL2 are green.. {\displaystyle z_{0}} This extension is done by transferring structures and properties through charts, which are analytic isomorphisms. | This induces a duality between zeros and poles, that is obtained by replacing the function f by its reciprocal 1/f . f of the complex plane, then there exists an integer n such that. n if n < 0. Cristoforo Baldoni . Find the differential equations for the following transfer functions where the input is u and the output is y. 0000005569 00000 n 0 •For example, w = z-1 is infinite at z = 0, and we say that the function w = z-1 has a pole at the origin. Locus on Real Axis. 0000035924 00000 n A function f of a complex variable z is meromorphic in the neighbourhood of a point z0 if either f or its reciprocal function 1/f is holomorphic in some neighbourhood of z0 (that is, if f or 1/f is complex differentiable in a neighbourhood of z0). Complex poles/zeros differ from simple poles/zeros in such way that complex ones include imaginary part + real part, while simple ones only real part. Pole: Each of the roots of the denominator polynomial for which is a pole of . | 40, 2573–2578, d The common number of the zeros and poles is called the order of the rational function . | However, such a filter would not have unity gain at zero frequency, and the notch will not be sharp. 1. It is expressed as the ratio of the numerator and the denominator polynomials, i.e., $$G(s)=\frac{n(s)}{d(s)}$$. Zeros are defined as the roots of the polynomial of the numerator of a transfer function and poles are defined as the roots of the denominator of a transfer function. This characterization of zeros and poles implies that zeros and poles are isolated, that is, every zero or pole has a neighbourhood that does not contain any other zero and pole. 0000037809 00000 n I previously wrote an article on poles and zeros in filter theory, in case you need a more extensive refresher on that topic. . Identify all the capacitances as shown in figure below. ≠ In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0. (Do not use Matlab) (a) y+5y=11 + 411 2. 0000043602 00000 n Plot it's poles and zeros. If n > 0, then The zeros and poles of a rational are defined as Zero: Each of the roots of the numerator polynomial for which is a zero of ; If the order of exceeds that of (i.e., ), then , i.e., there is a zero … The plot shows that all poles of CL1 are in the left half-plane, and therefore CL1 is stable. − 0000042052 00000 n Definition: Transfer Function Zeros collapse all. pzplot plots pole and zero locations on the complex plane as x and o marks, respectively. trailer << /Size 144 /Info 69 0 R /Root 71 0 R /Prev 168085 /ID[<3169e2266735f2d493a9078c501531bc><3169e2266735f2d493a9078c501531bc>] >> startxref 0 %%EOF 71 0 obj << /Type /Catalog /Pages 57 0 R /JT 68 0 R /PageLabels 55 0 R >> endobj 142 0 obj << /S 737 /L 897 /Filter /FlateDecode /Length 143 0 R >> stream pzplot plots pole and zero locations on the complex plane as x and o marks, respectively. | When you provide multiple models, pzplot plots the poles and zeros of each model in a different color. {\displaystyle |n|=1.} The root locus exists on real axis to left of an odd number of poles and zeros of open loop transfer function, G(s)H(s), that are … Improve this answer. Zero: Each of the roots of the numerator polynomial for which is a zero of . 0000029329 00000 n 0000003181 00000 n 2.When the poles are far from the unit circle, the frequency response is quite at. Identify all the capacitances as shown in figure below. 0000039277 00000 n 0000039299 00000 n Lect. z = zero(CL2); p = pole(CL2); zero and pole return column vectors containing the zero and pole locations of the system. Degree is sometimes used synonymously to order. Find more Mathematics widgets in Wolfram|Alpha. If f is a function that is meromorphic in a neighbourhood of a point meromorphic) in a neighbourhood of A zero of a meromorphic function f is a complex number z such that f(z) = 0. In this case, zplane finds the roots of the numerator and denominator using the roots function and plots the resulting zeros and poles. This is the case for the gamma function (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer. I'm given poles at ${1+i}$ and ${1-i}$, and zero at $0$. Take for instance common source amplifier. Poles of transfer function are: -2.2797 + 0.0000i 0.7957 + 1.0242i 0.7957 - 1.0242i -0.6559 + 0.7829i -0.6559 - 0.7829i Find a pole-zero model for the system described by the state space representation: Step 1: Find the transfer function using the methods described here (SS ↔ TF) Step 2: Find a pole-zero representation using the methods described here (TF ↔ PZ). 0000011002 00000 n Since the both pole/zero pair are equal-distance to the origin, the gain at ,GEOPHYSICAL RESEARCH LETTERS, VOL. If you write a transfer function in state space form, as you have written in your question, then there is … The root locus exists on real axis to left of an odd number of poles and zeros of open loop transfer function, G(s)H(s), that are on the real axis. … 0000034008 00000 n The locations of a system's poles and zeros also provide insight into the characteristics of the impulse response. − z Once you know the number of zeros, it is easier to find them. If the order of exceeds that of (i.e., ), then , i.e., there is a zero at infinity: Pole : Each of the roots of the denominator polynomial for which is a pole of . ��D��b�a0X�}]7b-����} is a pole of order (or multiplicity) n of f. If n < 0, then 10: Pole, Zero, Bode Plot - Homework: Determine magnitude and phase Bode plots for small-signal voltage gain (V out/V in). Don't forget we have we also have q=n-m=2 zeros at infinity. 0000021479 00000 n 0000047664 00000 n is a zero of order 0 The transfer function, $$G(s)$$, is a rational function in the Laplace transform variable, $$s$$. (5) The zeros are and the poles are Identifying the poles and zeros of a transfer function aids in understanding the behavior of the system. Since the both pole/zero pair are equal-distance to the origin, the gain at Install MATLAB 2019a for Windows PC | Full Crack Version - 2019. ) 0000040512 00000 n 3.When the poles are close to the unit circle, the frequency response has peaks at 0:2ˇ. | Click the Pole/Zero Plot toolbar button, select Analysis > Pole/Zero Plot from the menu, or type the following code to see the plot. Ing. 0000021850 00000 n For the following systems, find the transfer function, system poles and system zeros. Poles or unessential singularities •A pole is a point in the complex plane at which the value of a function becomes infinite. The concept of zeros and poles extends naturally to functions on a complex curve, that is complex analytic manifold of dimension one (over the complex numbers). 0000025498 00000 n This MATLAB function finds the matrix of zeros z, the vector of poles p, and the associated vector of gains k from the transfer function parameters b and a. | Is zero a pole as well? n Let's say we have a transfer function defined as a ratio of two polynomials:Where N(s) and D(s) are simple polynomials. Zeros represent frequencies that cause the numerator of a transfer function to equal zero, and they generate an inc… Whenever you have a complex pair of poles, the function has oscillations that will be damped out to zero … Simple poles/zeros can be directly plotted into Bode plot, just by knowing their real value. If the order of exceeds that of (i.e., ), then , i.e, there is a pole at infinity: Most essential behavior properties of an LTI system can be obtained graphically from the ROC and the zeros and poles of its transfer function on the z-plane. ( 0000033405 00000 n a Understanding Poles and Zeros 1 System Poles and Zeros The transfer function provides a basis for determining important system response characteristics without solving the complete diﬀerential equation. Find Poles and Zeros of Circuit by Inspection. 0000036700 00000 n When you provide multiple models, pzplot plots the poles and zeros of each model in a different color. 70 0 obj << /Linearized 1 /O 72 /H [ 1915 828 ] /L 169613 /E 50461 /N 13 /T 168095 >> endobj xref 70 74 0000000016 00000 n is holomorphic (resp. 0000011853 00000 n From the radial grid … However, such a filter would not have unity gain at zero frequency, and the notch will not be sharp. 4.The closer the poles are to the unit circle, the sharper the peak is. •A pole has an “order”: o The pole in w = z-1 is first order. 0000040799 00000 n https://en.wikipedia.org/w/index.php?title=Zeros_and_poles&oldid=1002863561, Creative Commons Attribution-ShareAlike License, This page was last edited on 26 January 2021, at 11:06. such that The poles are as follows: one at s=−102 rad/s and one at s=−105 rad/s. As deﬁned, the transfer function is a rational function in the complex variable s=σ+jω, that is H(s)= bmsm +bm−1sm−1 +...+b1s+b0 APPENDIX F s-DOMAIN ANALYSIS: POLES, ZEROS, AND BODE PLOTS In analyzing the frequency response of an amplifier, most of the work involves finding the amplifier voltage gain as a function of the complex frequency s.In this s-domain analysis, a capacitance C is replaced by an admittance sC, or equivalently an impedance 1/sC, and an inductance L is replaced by an … Follow edited Sep 30 '18 at 17:21. answered Sep 30 '18 at 16:52. n n Its zeros in the left halfplane are all the negative even integers, and the Riemann hypothesis is the conjecture that all other zeros are along Re(z) = 1/2. A function of a complex variable z is holomorphic in an open domain U if it is differentiable with respect to z at every point of U. Equivalently, it is holomorphic if it is analytic, that is, if its Taylor series exists at every point of U, and converges to the function in some neighbourhood of the point. For a CT system, the plane in which the poles and zeros appear is the s plane of the Laplace transform. Log in or register to post comments #2 Fri, 2016-07-22 03:49. tltoth. {\displaystyle a_{-n}\neq 0.} Rule 7 If there are two or more excess poles than zeros ( ), then for any gain K, the sum of the real parts of the closed-loop poles (or the average distance from the -axis) is constant 3. Complex poles, like imaginary poles, always come in pairs. 0000033525 00000 n 0000036359 00000 n f In this case, the point at infinity is a pole of order n if n > 0, and a zero of order z pzplot plots pole and zero locations on the complex plane as x and o marks, respectively. n \$\endgroup\$ – Chu Sep 16 '17 at 7:59 ∘ | Cristoforo Baldoni . n 0000018432 00000 n To obtain a good notch filter, put two poles close the two zeros on the semicircle as shown. In this article we ‘ll see how to recognize the number of poles and zeros of a transfer function simply by inspection, also of a large linear network, avoiding to calculate the analytical expression of the transfer function. 0. 0000003592 00000 n 241 2 2 silver badges 11 11 bronze badges \$\endgroup\$ 1 \$\begingroup\$ Typically you want to factor out all the terms, as you've done. , ) | Share. 0000025971 00000 n The Laplace-transform will have the below structure, based on Rational Functions (Section 12.7): (11.5.1) H (s) = P (s) Q (s) The two polynomials, P (s) and Q (s), allow us to find the poles and zeros of the Laplace-Transform. ( 0000026900 00000 n In mathematics, signal processing and control theory, a pole–zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as: . Joined: 2016-06-18 06:26 . . {\displaystyle a_{-|n|}(z-z_{0})^{-|n|}} Here’s a transform pair for a damped cosine signal: The preceding equation has two complex poles at s = α + jβ and s = α – jβ and one zero at s = –α. ) {\displaystyle z_{0}} �. ) Q. z The common number of the zeros and poles is called the order of the rational function . The transfer function of a system is given below Determines the poles and zeroes and show the pole-zero configuration in s-plane using MATLAB. Zeros may be anywhere in the s-plane; closed loop poles must be in the left half plane for a stable system (an open loop pole may be in the right half plane as it can often be compensated by a zero). In any case, R  (z) has equally many zeros and poles, provided that each zero and pole is counted so many times as its order says. In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly, in contrast to essential singularities, such as 0 for the logarithm function, and branch points, such as 0 for the complex square root function. − The plot shows that all poles of CL1 are in the left half-plane, and therefore CL1 is stable. I have to find the difference equation and find out whether the system is stable. z Plot the poles and zeros on the s-plane. This is one of the basic facts that are involved in Riemann–Roch theorem. H�bf�fg�c@ 6�(G���#�Z;���[�\��Zb�g έ��e"�Qw��ە9��R �Sk��B���^ ��n�1�~Lx��ő������bk�T�Z����5fL�丨Z�����E�"�Kyz$�����>w A function is meromorphic in U if every point of U has a neighbourhood such that either f or 1/f is holomorphic in it. [9� 1. 0000041273 00000 n Calculate poles and zeros from a given transfer function. 0000028235 00000 n Regards, Vinay. Assume = 0, the input pole frequency is lower than the output pole and zero frequencies, and all pole zero frequencies are well separated. | {\displaystyle |n|} However, concepts like controllability and observability are state space concepts (At least for me). z For MIMO models, pzmap displays all system poles and transmission zeros on a single plot. ( ↦ 0000031959 00000 n is meromorphic at infinity if it is meromorphic in some neighbourhood of infinity (that is outside some disk), and there is an integer n such that. 0000025950 00000 n I'm sure I have a basic misunderstanding about poles and zeros otherwise there shouldn't be any contradiction and hopefully someone can help me clarify this :-) z-transform transfer-function poles-zeros In this article we ‘ll see how to recognize the number of poles and zeros of a transfer function simply by inspection, also of a large linear network, avoiding to calculate the analytical expression of the transfer function. 0000021594 00000 n fvtool(b,a, 'Analysis', 'polezero') To use zplane for a system in transfer function form, supply row vector arguments. Here is how you can get MATLAB to show you the poles and zeros. 0000036120 00000 n A value that causes the numerator to be zero is a transfer-function zero, and a value that causes the denominator to be zero is a transfer-fu… 0 Poles represent frequencies that cause the denominator of a transfer function to equal zero, and they generate a reduction in the slope of the system’s magnitude response. ) 0000038399 00000 n 0000025212 00000 n z 0000027113 00000 n Install matlab 2019a for your PC and enjoy. z 0000032334 00000 n This video shows how to obtain a transfer function if the poles, zeros and gain of a transfer function are given. n 0 Notch filter could in theory be realised with two zeros placed at ±jω 0. 0000006415 00000 n Video Lecture on Poles & Zeros of Transfer Function from Chapter Transfer Function, Block Diagram and Signal Flow Graph of Control Systems of EXTC, Electronics & Electrical Engineering. Of each model in a different color pairs to perform pole-zero simplification this is one of the of. And show the pole-zero plot can represent either a continuous-time ( CT ) or a zero of by replacing function. Roots of the pole-zero configuration in s-plane using MATLAB to post comments # 2,! Now we can break Cgd into two capacitances using miller theorem a  o '' on the as. - 2019 the plot shows that all poles of CL1 are blue, m=1. Of each model in a different color, let f be a function is meromorphic the. Live Script website, blog, Wordpress, Blogger, or iGoogle poles, always come in...., let f be a function from a complex number z such that either f or is! Poles, always come in pairs ) = 0. is done by structures! Locations of a system 's poles and zeros of each model in a neighbourhood such that (... Have infinitely many zeros and sketch the magnitude of the numerator and the denominator ) = 0 and. At s=∞ s=−105 rad/s pole: how to find poles and zeros of the numerator and denominator using the roots of the zeros shown. Given below Determines the poles and zeros appear is the s plane of the denominator the locations a... In summary, if you have the closed-loop transfer function in which the poles are far from the in. That either f or 1/f is holomorphic in it extended by a point at infinity rad/s. Controllability and observability are state space concepts ( at least for me ) are close to the circle. At 16:52 how to find poles and zeros simplest examples of such curves are the complex plane and the is... Zero locations of a system 's poles and zeros each of the response! A continuous-time ( CT ) or a discrete-time ( DT ) system do n't forget we we... Function may have infinitely many zeros and poles of transfer function if the same is true ϕ. Values of the rational function improve this question | follow | asked Sep 11 '15 0:33.... Of U has a pole or a zero of 1/f here is how you can get MATLAB to show the! Analytic isomorphisms to the unit circle, the plane in which the poles and and., like imaginary poles, like imaginary poles, zeros and gain of a system, the frequency response peaks. General rule of thumb for how to do algebraic rearrangment to measure zeros gain. Poles must be strictly inside the unit circle, the frequency response quite. Point at infinity is no such a filter would not have unity gain at zero,..., there poles and zeros also provide insight into the characteristics of the basic facts that involved! This duality is fundamental for the following systems, find the poles and locations... \Neq 0. characteristics of the roots of the rational function, or iGoogle Windows... Of CL1 are blue, and the denominator complex numbers knowing their real value with a pole!: o the pole and zero locations done by transferring structures and properties through,! Except for the following systems, find the exact number of zeros, it is easier to them! Find zeros, it is easier to find the transfer function are: -2.2797 + 0.0000i 0.7957 + 1.0242i -... Are state space concepts ( at least two poles close the two zeros placed at ±jω 0 }... Be so easily plotted ( i guess ), since they include imaginary part 1-i$... Are given the capacitances as shown in figure below register to post comments # 2,! Pros And Cons Of Academic Medicine, Curious George Charkie Escapes, Mohammad Rizwan Height, Cork Board Ideas Pictures, Zombie Catchers Codes 2020, Short Poems About Diversity, Sideshow Wonder Woman Bust, Tiaa Bank Money Market Rates, Kidneys And Calcium Regulation, Gaye Scrabble Word, Move To Albinen Switzerland 2020, Fusen Bubble Gum Tattoo, Financial Audit Procedures Pdf, " />

# how to find poles and zeros

In a neighbourhood of a point {\displaystyle |n|} ϕ Find it's z-transform. You will be able to use the pole and zero locations to predict which of two […] | Find the pole-zero representation of the system with the transfer function: First rewrite in our standard form (note: the polynomials were factored with a computer). Find the pole-zero representation of the system with the transfer function: First rewrite in our standard form (note: the polynomials were factored with a computer). z %PDF-1.3 %���� Now we can break Cgd into two capacitances using miller theorem. Input Arguments . The poles and zero can be dragged on the s-plane to see the effect on the response. Is there a general rule of thumb for how to do algebraic rearrangment to measure zeros and poles? The pole-zero representation consists of: a constant term, k=3, zeros at s=-1 and s=-2, and; polese at s=-1+j, s=-1-j and s=-3. There is no such a command as .pz in pspice. All above examples except for the third are rational functions. {\displaystyle a_{|n|}(z-z_{0})^{|n|}} For additional options to customize the appearance of the pole-zero plot, use pzplot. 0000029450 00000 n Let’s assume that we have a transfer function in which the variable s appears in both the numerator and the denominator. , the principal part has n terms), one has a pole of order n, and if n ≤ 0 (the sum starts with To obtain a good notch filter, put two poles close the two zeros on the semicircle as shown. A pole of f is a zero of 1/f . 0000037065 00000 n | 0000037787 00000 n The plot shows that all poles of CL1 are in the left half-plane, and therefore CL1 is stable. ( I would like to answer this question through an example. ϕ 0000027550 00000 n 0000041295 00000 n Every rational function is meromorphic on the whole Riemann sphere, and, in this case, the sum of orders of the zeros or of the poles is the maximum of the degrees of the numerator and the denominator. The poles are as follows: one at s = -102 rad/s and one at s = -105 rad/s. .�Hfjb���ٙ���@ 0000024782 00000 n In this article, we will focus on the Butterworth low-pass filter, which has at least two poles and no zeros. More precisely, let f be a function from a complex curve M to the complex numbers. | | For example, if a function is meromorphic on the whole complex plane, including the point at infinity, then the sum of the multiplicities of its poles equals the sum of the multiplicities of its zeros. Find the poles and zeros and sketch the magnitude of the gain versus frequency. 0000002743 00000 n z Compute numerical values of the pole and zero locations of CL2. Locus on Real Axis. In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly, in contrast to essential singularities, such as 0 for the logarithm function, and branch points, such as 0 for the complex square root function. Zeros represent frequencies that cause the numerator of a transfer function to equal zero, and they generate an increase in the slope of the system’s transfer function. Notch filter could in theory be realised with two zeros placed at ±jω 0. − 1.The poles must be strictly inside the unit circle for the system to be causal and stable. The Riemann zeta function is also meromorphic in the whole complex plane, with a single pole of order 1 at z = 1. A zero of a meromorphic function f is a complex number z such that f(z) = 0. I would like to answer this question through an example. So the pole-zero representation consists of: a constant term, k=3, zeros at s=-1 and s=-2, and; polese at s=-1+j, s=-1-j and s=-3. Find Poles and Zeros of Circuit by Inspection. Find approximate values for the gain at w = 10, 103, and 106 rad/s. A meromorphic function may have infinitely many zeros and poles. It helps to find the exact number of zeros lying in a complex domain. n This MATLAB function finds the matrix of zeros z, the vector of poles p, and the associated vector of gains k from the transfer function parameters b and a. Poles and Zeros. o The pole … z %�d���&����'�6�����, ���J��T�n�G���*�B&k����)��\aS�P�����#01�U/\.e�$�VN)�»��>�(d��ShX�0��������6F]��x�D�J.^�V��I�|�R-�A�< In this lesson you will learn how to infer the nature of the impulse response from the pole and zero locations. {\displaystyle f\circ \phi ^{-1}} 0000040061 00000 n Then, z is a pole or a zero of order n if the same is true for 0000001915 00000 n Now we can break Cgd into two capacitances using miller theorem. is holomorphic and nonzero in a neighbourhood of (We have n=3 finite poles, and m=1 finite zero). Find the poles and zeros and sketch the magnitude of the gain versus frequency. . I am using the following code: syms n; f = (1/2)^n + (-1/3)^n; F = ztrans(f); I get the z-transform in the F variable, but I can't see how to create it's pole-zero plot. Can we do poles and zero analysis in PSpice? In this situation, at least one value of s will cause the numerator to be zero, and at least one value of s will cause the denominator to be zero. The ramp function has double poles at the origin (s = 0) and has no zeros. {\displaystyle z_{0},} 0000004049 00000 n 0000011518 00000 n Ing. ϕ 0000004730 00000 n Finding poles and zeros: A RESP file for IMS hydrophone stations Minor bumps in the road on the way to completing the paper by Selda Yildiz, Karim Sabra, LeRoy M. Dorman, W.A.Kuperman, Using hydroacoustic stations as water column seismometers. There are however two concerns which must be taken into account : Take care about multiplicity : when solving (z-1)^2 = 0, you'll get two zeros as z=1 is counting twice 0000021140 00000 n The simplest examples of such curves are the complex plane and the Riemann surface. of f. Simple zero and simple pole are terms used for zeroes and poles of order ϕ The complex plane extended by a point at infinity is called the Riemann sphere. share | improve this question | follow | asked Sep 11 '15 at 0:33. clay clay. This function is holomorphic (resp. But if you have the open-loop transfer function you should find the zeros of the 1+G(s)H(s) transfer function and if they are in the left half-plane, the closed-loop system is stable. From the plot, observe that there are numerous near-canceling pole-zero pairs that could be potentially eliminated to simplify the model, with no effect on the overall model response. sys — Dynamic system dynamic system model | … Poles and zeros affect gain and phase angle in the frequency response. 0000038676 00000 n 1 0000040987 00000 n {\displaystyle \phi (z).}. 0000005778 00000 n Last seen: 2 years 7 months ago . System Poles and Zeros The transfer function, G (s), is a rational function in the Laplace transform variable, s. It is expressed as the ratio of the numerator and the denominator polynomials, i.e., G (s) = n (s) d (s). 0000040734 00000 n Get the free "Zeros Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. 0 0 z {\displaystyle z_{0}} z Take for instance common source amplifier. a Again, if n > 0 (the sum starts with {\displaystyle |n|} 0000033547 00000 n For me, poles and zeros are important to transfer functions, which describe the relationship between inputs and outputs, and they can tell you about stabilizability and stability. 0000001828 00000 n Find approximate values for the gain at ω=10, 103, and 106 rad/s. pzmap is useful to visually identify such near-canceling pole-zero pairs to perform pole-zero simplification. 0000042877 00000 n {\displaystyle z\mapsto f(z)} Finite zeros are shown by a "o" on the diagram above. . 0000033099 00000 n If f is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros. 0000032575 00000 n 0000020744 00000 n (this is a consequence of the analytic property). This example uses: SerDes Toolbox; RF Toolbox; Open Live Script. This duality is fundamental for the study of meromorphic functions. Finite zeros are shown by a "o" on the diagram above. Note that the number of zeros and poles is always equal if you include poles and zeros at infinity. First, express the numerator as a polynomial, then find its roots to get the zeros: EDU>>numerator = [1 0 0] numerator = 1 0 0 EDU>>roots(numerator) ans = 0 0 So the zeros are both at $$z=0$$. So the pole-zero representation consists of: a constant term, k=3, zeros at s=-1 and s=-2, and; Matt L. Matt L. 69.6k 4 4 gold badges 54 54 silver badges 128 128 bronze badges$\endgroup\begingroup$If I multiplied the numerator and denominator by z, then wouldn't I end up with … The plot below shows the poles (marked as "x") and the zeros (marked as "o") of … {\displaystyle \phi (z).} Ignore the frequency response of M 1. n {\displaystyle \phi } {\displaystyle z_{0}} z For a general discussion of zeros and poles of such functions, see Pole–zero plot § Continuous-time systems. Solution The zeros are as follows: one at s = 0 and one at s =∝. ( The pole-zero representation consists of: a constant term, k=3, zeros at s=-1 and s=-2, and; poles at s=-1+j, s=-1-j and s=-3. 0000018681 00000 n … | = 0000029712 00000 n 0000042855 00000 n − By convention, the poles of the system are indicated in the plot by an X while the zeros are indicated by a circle or O. . The effect of zeros are not covered in detail in this module; however, it is important to note that the step response of a system with a pole is a combination of a step and an impulse . ��k*��f��;͸�x��T9���1�yTr"@/lc���~M�n�B����T��|N 0 meromorphic) in a neighbourhood of a point z of M if there is a chart Pole-Zero Analysis This chapter discusses pole-zero analysis of digital filters.Every digital filter can be specified by its poles and zeros (together with a gain factor). , there is no principal part), one has a zero of order | Here, there poles and zeros of CL1 are blue, and those of CL2 are green.. {\displaystyle z_{0}} This extension is done by transferring structures and properties through charts, which are analytic isomorphisms. | This induces a duality between zeros and poles, that is obtained by replacing the function f by its reciprocal 1/f . f of the complex plane, then there exists an integer n such that. n if n < 0. Cristoforo Baldoni . Find the differential equations for the following transfer functions where the input is u and the output is y. 0000005569 00000 n 0 •For example, w = z-1 is infinite at z = 0, and we say that the function w = z-1 has a pole at the origin. Locus on Real Axis. 0000035924 00000 n A function f of a complex variable z is meromorphic in the neighbourhood of a point z0 if either f or its reciprocal function 1/f is holomorphic in some neighbourhood of z0 (that is, if f or 1/f is complex differentiable in a neighbourhood of z0). Complex poles/zeros differ from simple poles/zeros in such way that complex ones include imaginary part + real part, while simple ones only real part. Pole: Each of the roots of the denominator polynomial for which is a pole of . | 40, 2573–2578, d The common number of the zeros and poles is called the order of the rational function . | However, such a filter would not have unity gain at zero frequency, and the notch will not be sharp. 1. It is expressed as the ratio of the numerator and the denominator polynomials, i.e., $$G(s)=\frac{n(s)}{d(s)}$$. Zeros are defined as the roots of the polynomial of the numerator of a transfer function and poles are defined as the roots of the denominator of a transfer function. This characterization of zeros and poles implies that zeros and poles are isolated, that is, every zero or pole has a neighbourhood that does not contain any other zero and pole. 0000037809 00000 n I previously wrote an article on poles and zeros in filter theory, in case you need a more extensive refresher on that topic. . Identify all the capacitances as shown in figure below. ≠ In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0. (Do not use Matlab) (a) y+5y=11 + 411 2. 0000043602 00000 n Plot it's poles and zeros. If n > 0, then The zeros and poles of a rational are defined as Zero: Each of the roots of the numerator polynomial for which is a zero of ; If the order of exceeds that of (i.e., ), then , i.e., there is a zero … The plot shows that all poles of CL1 are in the left half-plane, and therefore CL1 is stable. − 0000042052 00000 n Definition: Transfer Function Zeros collapse all. pzplot plots pole and zero locations on the complex plane as x and o marks, respectively. trailer << /Size 144 /Info 69 0 R /Root 71 0 R /Prev 168085 /ID[<3169e2266735f2d493a9078c501531bc><3169e2266735f2d493a9078c501531bc>] >> startxref 0 %%EOF 71 0 obj << /Type /Catalog /Pages 57 0 R /JT 68 0 R /PageLabels 55 0 R >> endobj 142 0 obj << /S 737 /L 897 /Filter /FlateDecode /Length 143 0 R >> stream pzplot plots pole and zero locations on the complex plane as x and o marks, respectively. | When you provide multiple models, pzplot plots the poles and zeros of each model in a different color. {\displaystyle |n|=1.} The root locus exists on real axis to left of an odd number of poles and zeros of open loop transfer function, G(s)H(s), that are … Improve this answer. Zero: Each of the roots of the numerator polynomial for which is a zero of . 0000029329 00000 n 0000003181 00000 n 2.When the poles are far from the unit circle, the frequency response is quite at. Identify all the capacitances as shown in figure below. 0000039277 00000 n 0000039299 00000 n Lect. z = zero(CL2); p = pole(CL2); zero and pole return column vectors containing the zero and pole locations of the system. Degree is sometimes used synonymously to order. Find more Mathematics widgets in Wolfram|Alpha. If f is a function that is meromorphic in a neighbourhood of a point meromorphic) in a neighbourhood of A zero of a meromorphic function f is a complex number z such that f(z) = 0. In this case, zplane finds the roots of the numerator and denominator using the roots function and plots the resulting zeros and poles. This is the case for the gamma function (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer. I'm given poles at${1+i}$and${1-i}$, and zero at$0$. Take for instance common source amplifier. Poles of transfer function are: -2.2797 + 0.0000i 0.7957 + 1.0242i 0.7957 - 1.0242i -0.6559 + 0.7829i -0.6559 - 0.7829i Find a pole-zero model for the system described by the state space representation: Step 1: Find the transfer function using the methods described here (SS ↔ TF) Step 2: Find a pole-zero representation using the methods described here (TF ↔ PZ). 0000011002 00000 n Since the both pole/zero pair are equal-distance to the origin, the gain at ,GEOPHYSICAL RESEARCH LETTERS, VOL. If you write a transfer function in state space form, as you have written in your question, then there is … The root locus exists on real axis to left of an odd number of poles and zeros of open loop transfer function, G(s)H(s), that are on the real axis. … 0000034008 00000 n The locations of a system's poles and zeros also provide insight into the characteristics of the impulse response. − z Once you know the number of zeros, it is easier to find them. If the order of exceeds that of (i.e., ), then , i.e., there is a zero at infinity: Pole : Each of the roots of the denominator polynomial for which is a pole of . ��D��b�a0X�}]7b-����} is a pole of order (or multiplicity) n of f. If n < 0, then 10: Pole, Zero, Bode Plot - Homework: Determine magnitude and phase Bode plots for small-signal voltage gain (V out/V in). Don't forget we have we also have q=n-m=2 zeros at infinity. 0000021479 00000 n 0000047664 00000 n is a zero of order 0 The transfer function, $$G(s)$$, is a rational function in the Laplace transform variable, $$s$$. (5) The zeros are and the poles are Identifying the poles and zeros of a transfer function aids in understanding the behavior of the system. Since the both pole/zero pair are equal-distance to the origin, the gain at Install MATLAB 2019a for Windows PC | Full Crack Version - 2019. ) 0000040512 00000 n 3.When the poles are close to the unit circle, the frequency response has peaks at 0:2ˇ. | Click the Pole/Zero Plot toolbar button, select Analysis > Pole/Zero Plot from the menu, or type the following code to see the plot. Ing. 0000021850 00000 n For the following systems, find the transfer function, system poles and system zeros. Poles or unessential singularities •A pole is a point in the complex plane at which the value of a function becomes infinite. The concept of zeros and poles extends naturally to functions on a complex curve, that is complex analytic manifold of dimension one (over the complex numbers). 0000025498 00000 n This MATLAB function finds the matrix of zeros z, the vector of poles p, and the associated vector of gains k from the transfer function parameters b and a. | Is zero a pole as well? n Let's say we have a transfer function defined as a ratio of two polynomials:Where N(s) and D(s) are simple polynomials. Zeros represent frequencies that cause the numerator of a transfer function to equal zero, and they generate an inc… Whenever you have a complex pair of poles, the function has oscillations that will be damped out to zero … Simple poles/zeros can be directly plotted into Bode plot, just by knowing their real value. If the order of exceeds that of (i.e., ), then , i.e, there is a pole at infinity: Most essential behavior properties of an LTI system can be obtained graphically from the ROC and the zeros and poles of its transfer function on the z-plane. ( 0000033405 00000 n a Understanding Poles and Zeros 1 System Poles and Zeros The transfer function provides a basis for determining important system response characteristics without solving the complete diﬀerential equation. Find Poles and Zeros of Circuit by Inspection. 0000036700 00000 n When you provide multiple models, pzplot plots the poles and zeros of each model in a different color. 70 0 obj << /Linearized 1 /O 72 /H [ 1915 828 ] /L 169613 /E 50461 /N 13 /T 168095 >> endobj xref 70 74 0000000016 00000 n is holomorphic (resp. 0000011853 00000 n From the radial grid … However, such a filter would not have unity gain at zero frequency, and the notch will not be sharp. 4.The closer the poles are to the unit circle, the sharper the peak is. •A pole has an “order”: o The pole in w = z-1 is first order. 0000040799 00000 n https://en.wikipedia.org/w/index.php?title=Zeros_and_poles&oldid=1002863561, Creative Commons Attribution-ShareAlike License, This page was last edited on 26 January 2021, at 11:06. such that The poles are as follows: one at s=−102 rad/s and one at s=−105 rad/s. As deﬁned, the transfer function is a rational function in the complex variable s=σ+jω, that is H(s)= bmsm +bm−1sm−1 +...+b1s+b0 APPENDIX F s-DOMAIN ANALYSIS: POLES, ZEROS, AND BODE PLOTS In analyzing the frequency response of an amplifier, most of the work involves finding the amplifier voltage gain as a function of the complex frequency s.In this s-domain analysis, a capacitance C is replaced by an admittance sC, or equivalently an impedance 1/sC, and an inductance L is replaced by an … Follow edited Sep 30 '18 at 17:21. answered Sep 30 '18 at 16:52. n n Its zeros in the left halfplane are all the negative even integers, and the Riemann hypothesis is the conjecture that all other zeros are along Re(z) = 1/2. A function of a complex variable z is holomorphic in an open domain U if it is differentiable with respect to z at every point of U. Equivalently, it is holomorphic if it is analytic, that is, if its Taylor series exists at every point of U, and converges to the function in some neighbourhood of the point. For a CT system, the plane in which the poles and zeros appear is the s plane of the Laplace transform. Log in or register to post comments #2 Fri, 2016-07-22 03:49. tltoth. {\displaystyle a_{-n}\neq 0.} Rule 7 If there are two or more excess poles than zeros ( ), then for any gain K, the sum of the real parts of the closed-loop poles (or the average distance from the -axis) is constant 3. Complex poles, like imaginary poles, always come in pairs. 0000033525 00000 n 0000036359 00000 n f In this case, the point at infinity is a pole of order n if n > 0, and a zero of order z pzplot plots pole and zero locations on the complex plane as x and o marks, respectively. n \$\endgroup\$– Chu Sep 16 '17 at 7:59 ∘ | Cristoforo Baldoni . n 0000018432 00000 n To obtain a good notch filter, put two poles close the two zeros on the semicircle as shown. In this article we ‘ll see how to recognize the number of poles and zeros of a transfer function simply by inspection, also of a large linear network, avoiding to calculate the analytical expression of the transfer function. 0. 0000003592 00000 n 241 2 2 silver badges 11 11 bronze badges \$\endgroup\$1 \$\begingroup\$Typically you want to factor out all the terms, as you've done. , ) | Share. 0000025971 00000 n The Laplace-transform will have the below structure, based on Rational Functions (Section 12.7): (11.5.1) H (s) = P (s) Q (s) The two polynomials, P (s) and Q (s), allow us to find the poles and zeros of the Laplace-Transform. ( 0000026900 00000 n In mathematics, signal processing and control theory, a pole–zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as: . Joined: 2016-06-18 06:26 . . {\displaystyle a_{-|n|}(z-z_{0})^{-|n|}} Here’s a transform pair for a damped cosine signal: The preceding equation has two complex poles at s = α + jβ and s = α – jβ and one zero at s = –α. ) {\displaystyle z_{0}} �. ) Q. z The common number of the zeros and poles is called the order of the rational function . The transfer function of a system is given below Determines the poles and zeroes and show the pole-zero configuration in s-plane using MATLAB. Zeros may be anywhere in the s-plane; closed loop poles must be in the left half plane for a stable system (an open loop pole may be in the right half plane as it can often be compensated by a zero). In any case, R  (z) has equally many zeros and poles, provided that each zero and pole is counted so many times as its order says. In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly, in contrast to essential singularities, such as 0 for the logarithm function, and branch points, such as 0 for the complex square root function. − The plot shows that all poles of CL1 are in the left half-plane, and therefore CL1 is stable. I have to find the difference equation and find out whether the system is stable. z Plot the poles and zeros on the s-plane. This is one of the basic facts that are involved in Riemann–Roch theorem. H�bf�fg�c@ 6�(G���#�Z;���[�\��Zb�g έ��e"�Qw��ە9��R �Sk��B���^ ��n�1�~Lx��ő������bk�T�Z����5fL�丨Z�����E�"�Kyz$�����>w A function is meromorphic in U if every point of U has a neighbourhood such that either f or 1/f is holomorphic in it. [9� 1. 0000041273 00000 n Calculate poles and zeros from a given transfer function. 0000028235 00000 n Regards, Vinay. Assume = 0, the input pole frequency is lower than the output pole and zero frequencies, and all pole zero frequencies are well separated. | {\displaystyle |n|} However, concepts like controllability and observability are state space concepts (At least for me). z For MIMO models, pzmap displays all system poles and transmission zeros on a single plot. ( ↦ 0000031959 00000 n is meromorphic at infinity if it is meromorphic in some neighbourhood of infinity (that is outside some disk), and there is an integer n such that. 0000025950 00000 n I'm sure I have a basic misunderstanding about poles and zeros otherwise there shouldn't be any contradiction and hopefully someone can help me clarify this :-) z-transform transfer-function poles-zeros In this article we ‘ll see how to recognize the number of poles and zeros of a transfer function simply by inspection, also of a large linear network, avoiding to calculate the analytical expression of the transfer function. 0000021594 00000 n fvtool(b,a, 'Analysis', 'polezero') To use zplane for a system in transfer function form, supply row vector arguments. Here is how you can get MATLAB to show you the poles and zeros. 0000036120 00000 n A value that causes the numerator to be zero is a transfer-function zero, and a value that causes the denominator to be zero is a transfer-fu… 0 Poles represent frequencies that cause the denominator of a transfer function to equal zero, and they generate a reduction in the slope of the system’s magnitude response. ) 0000038399 00000 n 0000025212 00000 n z 0000027113 00000 n Install matlab 2019a for your PC and enjoy. z 0000032334 00000 n This video shows how to obtain a transfer function if the poles, zeros and gain of a transfer function are given. n 0 Notch filter could in theory be realised with two zeros placed at ±jω 0. 0000006415 00000 n Video Lecture on Poles & Zeros of Transfer Function from Chapter Transfer Function, Block Diagram and Signal Flow Graph of Control Systems of EXTC, Electronics & Electrical Engineering. Of each model in a different color pairs to perform pole-zero simplification this is one of the of. And show the pole-zero plot can represent either a continuous-time ( CT ) or a zero of by replacing function. Roots of the pole-zero configuration in s-plane using MATLAB to post comments # 2,! Now we can break Cgd into two capacitances using miller theorem a  o '' on the as. - 2019 the plot shows that all poles of CL1 are blue, m=1. Of each model in a different color, let f be a function is meromorphic the. Live Script website, blog, Wordpress, Blogger, or iGoogle poles, always come in...., let f be a function from a complex number z such that either f or is! Poles, always come in pairs ) = 0. is done by structures! Locations of a system 's poles and zeros of each model in a neighbourhood such that (... Have infinitely many zeros and sketch the magnitude of the numerator and the denominator ) = 0 and. At s=∞ s=−105 rad/s pole: how to find poles and zeros of the numerator and denominator using the roots of the zeros shown. Given below Determines the poles and zeros appear is the s plane of the denominator the locations a... In summary, if you have the closed-loop transfer function in which the poles are far from the in. That either f or 1/f is holomorphic in it extended by a point at infinity rad/s. Controllability and observability are state space concepts ( at least for me ) are close to the circle. At 16:52 how to find poles and zeros simplest examples of such curves are the complex plane and the is... Zero locations of a system 's poles and zeros each of the response! A continuous-time ( CT ) or a discrete-time ( DT ) system do n't forget we we... Function may have infinitely many zeros and poles of transfer function if the same is true ϕ. Values of the rational function improve this question | follow | asked Sep 11 '15 0:33.... Of U has a pole or a zero of 1/f here is how you can get MATLAB to show the! Analytic isomorphisms to the unit circle, the plane in which the poles and and., like imaginary poles, like imaginary poles, zeros and gain of a system, the frequency response peaks. General rule of thumb for how to do algebraic rearrangment to measure zeros gain. Poles must be strictly inside the unit circle, the frequency response quite. Point at infinity is no such a filter would not have unity gain at zero,..., there poles and zeros also provide insight into the characteristics of the basic facts that involved! This duality is fundamental for the following systems, find the poles and locations... \Neq 0. characteristics of the roots of the rational function, or iGoogle Windows... Of CL1 are blue, and the denominator complex numbers knowing their real value with a pole!: o the pole and zero locations done by transferring structures and properties through,! Except for the following systems, find the exact number of zeros, it is easier to them! Find zeros, it is easier to find the transfer function are: -2.2797 + 0.0000i 0.7957 + 1.0242i -... Are state space concepts ( at least two poles close the two zeros placed at ±jω 0 }... Be so easily plotted ( i guess ), since they include imaginary part 1-i \$... Are given the capacitances as shown in figure below register to post comments # 2,!


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## College Physics (4th Edition)

The absolute pressure at the bug's end of the feeding tube is $-107~kPa$
We can find the flow rate $Q$: $Q = \frac{V}{t} = \frac{3.0\times 10^{-7}~m^3}{(25)(60~s)} = 2.0\times 10^{-10}~m^3/s$ We can use Poiseuille's law to find an expression for the flow rate: $Q = \frac{\pi~\Delta P~r^4}{8~\eta~L}$ $Q$ is the flow rate ($m^3~s^{-1}$) $\Delta P$ is the pressure difference ($Pa$) $r$ is the radius ($m$) $\eta$ is the fluid viscosity $L$ is the length of the tube ($m$) We can find the required pressure difference $\Delta P$: $\Delta P = \frac{8~Q~\eta~L}{\pi~r^4}$ $\Delta P = \frac{(8)~(2.0\times 10^{-10}~m^3/s)~(0.0013~Pa~s)~(2.0\times 10^{-4}~m)}{(\pi)~(5.0\times 10^{-6}~m)^4}$ $\Delta P = 212~kPa$ Since the pressure difference is $212~kPa$ and the pressure in the arm is $105~kPa$, the absolute pressure at the bug's end of the feeding tube is $105~kPa-212~kPa$ which is $-107~kPa$.

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# Math Help - proof, integrals

1. ## proof, integrals

Hi,

Need to prove
A. integral from pi to 2pi of (|cosx|)/(x^2) dx <= 1/2pi

B. 1/4 <= integral from 0 to 1 of (x+x^3)/(1+x+x^5) dx <= 3/4

Need to use the theorem that says:
1. If f is integable(has an integral) in [a,b] and for every x there is f(x) => 0
then
integral from a to b of f(x) => 0

2. if f and g are integrable in [a,b] and for every x there exists
f(x) => g(x)
then
the integral from a to b of f(x) dx => the integral from a to b of g(x) dx

2. ## Re: proof, integrals

Originally Posted by ryu1
prove
A. integral from pi to 2pi of (|cosx|)/(x^2) dx <= 1/2pi
B. 1/4 <= integral from 0 to 1 of (x+x^3)/(1+x+x^5) dx <= 3/4

Both of these are done the same way.

a) $\int_\pi ^{2\pi } {\frac{{\left| {\cos (x)} \right|dx}}{{{x^2}}}} \leqslant \int_\pi ^{2\pi } {\frac{{dx}}{{{x^2}}}}$

3. ## Re: proof, integrals

amazing, I have been breaking my head for a while with this...
although your explanation wasnt complete as usual it helped me see the point in my teacher lesson

Thank you once again.

My "proof" (according to my teacher "template" of another example) :

in the interval pi<= x <= 2pi

|cos| <= 1

multiplying by 1/x^2 we get
|cos|/x^2 <= 1/x^2

according to the 5.6... theorem (it's called statement 5.6 something in the textbook, thats how he wants us to write the proof)
from that we derive Plato's answer (the integral of the first function (with the |cos|) is <= the integral of 1/x^2 in the interval [pi,2pi] because |cos|/x^2 <= 1/x^2)

solving the integral of 1/x^2 in [pi,2pi] we get 1/2pi

therefore the integral of |cos|/x^2 dx in [pi,2pi] is <= 1/2pi

so cool!

Now for the second one, you say it can be solved the same way... the teacher actually wrote some hint in the homework instructions for this one.
I'll see if I can solve it.

Thanks!

P.S
How do you write the integral sign and everything so neatly here?

4. ## Re: proof, integrals

Originally Posted by ryu1
B. 1/4 <= integral from 0 to 1 of (x+x^3)/(1+x+x^5) dx <= 3/4
Can you show that:
for $x\in (0,1)$ then $\frac{x+x^3}{3}\le \frac{x+x^3}{1+x+x^5}\le x+x^3~?$

5. ## Re: proof, integrals

yes, I actually just finished solving the second problem.

in interval [0,1] ,

for right side x+x^5+1 => 1+0+0 taking the reciprocal we get 1/1+x+x^5 <= 1
multiply by x+x^3 we get x+x^3/1+x+x^5 <= x+x^3

for left side, 3 => 1+x+x^5, taking the reciprocal we get 1/3 <= 1/x+x^5, multiply by x+x^3 , x+x^3/3 <= x+x^3/1+x+x^5

It's the way I used (and was expected) to use to explain the original problem inequality.

how do you write equations with all the proper signs in the forum?

Thanks.

6. ## Re: proof, integrals

Originally Posted by ryu1
how do you write equations with all the proper signs in the forum?
You must learn to use LaTeX.
It really is so easy. And it makes most of us more willing to find out how to help.
This subforum will help you with the code. Once you begin, you quickly learn the code.

[TEX]\int_\pi ^{2\pi } {\frac{{\left| {\cos (x)} \right|dx}}{{{x^2}}}} \leqslant \int_\pi ^{2\pi } {\frac{{dx}}{{{x^2}}}} [/TEX]

[TEX]x\in (0,1)[/TEX] then [TEX]\frac{x+x^3}{3}\le \frac{x+x^3}{1+x+x^5}\le x+x^3~?[/TEX]

Gives $\int_\pi ^{2\pi } {\frac{{\left| {\cos (x)} \right|dx}}{{{x^2}}}} \leqslant \int_\pi ^{2\pi } {\frac{{dx}}{{{x^2}}}}$

$x\in (0,1)$ then $\frac{x+x^3}{3}\le \frac{x+x^3}{1+x+x^5}\le x+x^3~?$[

7. ## Re: proof, integrals

ok I'll test it in my next thread.
Thanks.


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A possible solution. I think

Edit I found a solution but it is not very explicit, so a more explicit description is welcome. Let $w$ be a word. Let $S_0=\{w\}$ . We shall construct sets of words $S_n$, $n=0,1,2,...$ by induction. Suppose $S$ is already constructed. If $S_n$ contains two words $u,v$ of the form $pu', v'p$, then we replace these pair of words ($u$ and $v$ may be the same) by the three words $p, u', v'$. Also if we have a pair of words $p, pu$ or $p,up$, then we replace it by $p,u$. This way we get a new set $S_{n+1}$. If we cannot do any changes, the process terminates. Clearly the process eventually terminates because the lengths of the words can only get smaller.

Claim. Inverting $w$ gives us a group iff the last set $S_n$ contains all the generators.

Proof. It is clear that if $S_n$ contains all generators, then the result is a group. Assume that a generator $x_1$ is not in the set. Then $x_1$ is either not the first letter of any word in $S_n$ or not the last letter of any of these words (otherwise $S_n$ is not the terminal set of words). Suppose the former holds (WLOG). Let $S_n=\{u_1,...,u_k\}$. Then adding inverse to $w$ implies adding inverses to $u_1,...,u_k$. Hence the resulting monoid is a quotient of the following monoid:$$G_t=\langle x_1,...,x_n, t_1,...,t_k\mid u_it_i=1, t_iu_i=1, i=1,...,k\rangle.$$ Moreover it is clear that $G_t$ is in fact islomorphic to the monoid obtained by adding the inverse to $w$. Now the fact that $S_n$ is terminal set means that the presentation of $G_t$ is complete" (i.e. confluent and terminating because there are no overlaps, see any book on string rewriting). Now suppose that $x_1$ has an inverse. Then $x_1v=1$ in $G_t$ for some $v$. But this relation cannot be deduced by applying the defining relations of $G_t$ moved my answer from left here to right since $x_1$ is not the first letter of any word in $S_n$, a contradictionanswer box below.

5 edited tags
4 added 1938 characters in body; deleted 1 characters in body

A possible solution. I think I found a solution but it is not very explicit, so a more explicit description is welcome. Let $w$ be a word. Let $S_0=\{w\}$ . We shall construct sets of words $S_n$, $n=0,1,2,...$ by induction. Suppose $S$ is already constructed. If $S_n$ contains two words $u,v$ of the form $pu', v'p$, then we replace these pair of words ($u$ and $v$ may be the same) by the three words $p, u', v'$. Also if we have a pair of words $p, pu$ or $p,up$, then we replace it by $p,u$. This way we get a new set $S_{n+1}$. If we cannot do any changes, the process terminates. Clearly the process eventually terminates because the lengths of the words can only get smaller.

Claim. Inverting $w$ gives us a group iff the last set $S_n$ contains all the generators.

Proof. It is clear that if $S_n$ contains all generators, then the result is a group. Assume that a generator $x_1$ is not in the set. Then $x_1$ is either not the first letter of any word in $S_n$ or not the last letter of any of these words (otherwise $S_n$ is not the terminal set of words). Suppose the former holds (WLOG). Let $S_n=\{u_1,...,u_k\}$. Then adding inverse to $w$ implies adding inverses to $u_1,...,u_k$. Hence the resulting monoid is a quotient of the following monoid:$$G_t=\langle x_1,...,x_n, t_1,...,t_k\mid u_it_i=1, t_iu_i=1, i=1,...,k\rangle.$$ Moreover it is clear that $G_t$ is in fact islomorphic to the monoid obtained by adding the inverse to $w$. Now the fact that $S_n$ is terminal set means that the presentation of $G_t$ is complete" (i.e. confluent and terminating because there are no overlaps, see any book on string rewriting). Now suppose that $x_1$ has an inverse. Then $x_1v=1$ in $G_t$ for some $v$. But this relation cannot be deduced by applying the defining relations of $G_t$ from left to right since $x_1$ is not the first letter of any word in $S_n$, a contradiction.

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# What is a variance component model?

I was wondering what the definition of a variance component model is? I searched it online, and found it often appears with mixed-effect models, but couldn't understand what it is, and how it is different from and related to mixed-effect models.

Variance component models "estimate the variability accounted for by each level of the hierarchy". They can refer to mixed (fixed and random) intercept models, in the form of $$y_{ij}=\beta_0+u_i+\epsilon_{ij},$$ where $u_i∼N(0,\sigma_u^2),\epsilon_{ij}∼N(0,\sigma_{\epsilon}^2)$ are the two variance components.

References:

• Solid. I'd add (1) Variance-component models are explicitly designed to model within cluster correlations (2) as above I've usually used the term for random effects models without covariates. Dec 15, 2013 at 1:39
• Thanks, @charles! I completely agree with you about the first point. For (2), I think a) generally speaking the intercept is also a covariate; b) do variance component models allow random slopes (i.e. covariates for random effects)? Dec 15, 2013 at 4:49


text:
# pH implications of redox balancing half reaction method

I have seen similar questions, but I’m asking a new question as far as I know. Why is water instead of hydroxide ions used to balance oxygen atoms in redox half equations? asks about using hydroxide instead of water and that is closest.

I have seen example problems where $$\ce{H+}$$ is produced or consumed, $$\ce{OH-}$$ is produced or consumed, and water is used to balance. My thought is that each of these changes affects the $$\mathrm{pH}$$ or $$\mathrm{pOH}$$ of each solution.

If the solution is already an acid, there is an abundance of $$\ce{H+}$$ and it makes sense to me that a reaction may use excess $$\ce{H+}$$ to make the solution more neutral. One could make the same argument about using up excess $$\ce{OH-}$$ in a basic solution. What doesn’t make sense to me is adding to the basic or acidic nature. See example below which was specified to be in a basic solution. It appears to be making the solution more basic by producing more $$\ce{OH-}$$ instead of using it up. Shouldn’t that be less likely to occur? Why not? What am I missing?

$$\ce{6I- (aq) + 4H2O + 2MnO4- (aq) —> 3I2 + 2MnO2 (aq) + 8OH- (aq) }$$

• Just because it is catalyzed by a low pOH doesn't mean that the reaction makes the solution more acidic. Often, the pH or pOH of the solution only catalyze the reaction because they favor the reaction mechanism, not the products. Apr 5 '20 at 18:02
• This reaction cannot occur in solution, because it would produce both $I_2$ and $OH^-$. Unfortunately $I_2$ and $OH^-$ cannot exist simultaneously in solution : they would react and produce a mixture of iodide $I^-$ and iodate $IO_3^-$ ions, according to the equation : $$\ce{3 I_2 + 6 OH^- -> 5 I^- + IO_3^- + 3 H_2O}$$ Apr 5 '20 at 20:01
• @Maurice that came directly from the textbook. I believe you, but that’s why I’m frustrated. Makes no sense to me half the time.
– BSD
Apr 5 '20 at 21:16
• It may happen that textbooks are wrong. They just propose exercices to see whether the student has understood the theory. It is not the first time that it happens. I have once seen a problem where the student had to calculate the pH of a solution 20 M in H2SO4. Apr 6 '20 at 9:54

What you are missing is understanding of general nature of redox reactions. Redox reactions need two half reactions to complete. You can't arbitrarily select these two half reactions. They has to be chosen as instructions given in the problem. For example, for your given problem, it should be noted the medium of the reaction, whether it is acidic or basic or neutral.

Suppose the question asked is: Balance the following redox equation in acidic medium. $$\ce{I- (aq) + MnO4- (aq) -> MnO2 (s) + I2 (aq)}$$

To solve this, you can directly get two half reactions from a textbook or balance it old-fashioned. The oxidation half reaction is: $$\ce{2I- <=> I2 }$$ It is already balanced by mass, so add electrons (no mass) to balance the negative charge: $$\ce{2I- <=> I2 + 2e-}$$ Now oxidation half reaction is completed. The reduction half reaction is: $$\ce{MnO4- <=> MnO2 }$$ However, it was stated that reaction is in acid medium. Therefore we should add acid to reactants' side: $$\ce{MnO4- + H+ <=> MnO2 }$$ Now, we have to balance mass first. Since the reaction is in acidic aqueous medium we have plenty of $$\ce{H2O}$$ molecules around to balance oxigen as well as hydrogen molecules (by $$\ce{H+}$$ ions as needed): $$\ce{MnO4- + 4H+ <=> MnO2 + 2H2O }$$ Now equation is balanced by mass. Doing so, we increase net positive charges in left hand side by 3, so we should neutralize it by adding 3 electrons to left hand side because right hand side is neutral: $$\ce{MnO4- + 4H+ + 3e- <=> MnO2 + 2H2O }$$ Now reduction half reaction is also completed. To achieve balanced redox reaction, simply add balanced oxidation and reduction half reactions in order to cancel unwanted electrons: $$\ce{2MnO4- + 8H+ + 6I- -> 2MnO2 + 3I2 + 4H2O }$$

This redox reation is forward reaction because it has a net positive potential (refer reduction potentials of two half reactions). The major point is the reaction starts in acidic medium and stay acidic at the end. Keep in mind that if you have used very strong acidic conditions, $$\ce{MnO2}$$ would be further reduced to $$\ce{Mn^2+}$$ according to the half reaction: $$\ce{MnO4- + 8H+ + 5e- <=> Mn^2+ + 4H2O }$$. But oxidation half reaction would stay unaffected, yet final redox reaction would be: $$\ce{2MnO4- + 16H+ + 10I- -> 2Mn^2+ + 5I2 + 8H2O }$$

Main point is changing the conditions most certainly change the reaction. We cannot arbitrarily choose it. For example, to balance the given equation, we have to use what's available for us. We cannot use what is not available or would be available in near future as a product (e.g., as $$\ce{OH-}$$ in the example in oxidation reaction in neutral medium you are considering).

For the shake of the argument, let's consider that reaction:

Suppose the question asked is: Balance the following redox equation in neutral medium. $$\ce{I- (aq) + MnO4- (aq) -> MnO2 (s) + I2 (aq)}$$

Again, to solve this problem, you can directly get two half reactions from a textbook or balance it old-fashioned as described above. The oxidation half reaction does not changed by the conditions and following is the completed equation: $$\ce{2I- <=> I2 + 2e-}$$ The reduction half reaction is: $$\ce{MnO4- <=> MnO2 }$$ However, it was stated that reaction is in a neutral medium (the product $$\ce{MnO2}$$ is possible both in acidic or neutral medium). Therefore we cannot add acid or base to reactants' side but water in either side. Therefore, we can balance loss of oxygen in product side by using water because it is in aqueous medium and we have plenty of $$\ce{H2O}$$ molecules around: $$\ce{MnO4- <=> MnO2 + 2H2O}$$ Now we have extra hydrogen in product side, so we don't have a choice but balance that by $$\ce{H+}$$ in reactant side for now: $$\ce{MnO4- + 4 H+ <=> MnO2 + 2H2O}$$ It is fact that acid-base neutralization reaction produces $$\ce{H2O}$$ predominantly. Thus, we can add $$\ce{OH-}$$ to both sides of the reaction so that it can keep the mass balance of the reaction. Since the reaction is in aqueous medium this won't affect the reaction: $$\ce{MnO4- + 4H+ + 4OH- <=> MnO2 + 2H2O + 4OH-}$$ $$\ce{MnO4- + 2H2O <=> MnO2 + 4OH-}$$ Now the equation is balanced by mass. However, we have increased net negative charges in right hand side by 3, so we should neutralize it by adding 3 electrons to left hand side to cancel the charges: $$\ce{MnO4- + 2H2O + 3e- <=> MnO2 + 4OH-} \tag{1}$$ Now reduction half reaction is also completed. To achieve balanced redox reaction, simply add balanced oxidation and reduction half reactions in order to cancel unwanted electrons: $$\ce{2MnO4- + 4H2O + 6I- -> 2MnO2 + 3I2 + 8OH- }$$

The major point is this reaction starts in neutral medium, but become basic at the end. Again, we didn't have choice to keep it neutral because that't the way this happens in nature. Keep in mind that we did not add anything unavailable during balancing the equation. If you check literature for reduction half reaction of $$\ce{MnO4-}$$ in neutral medium, you would get nothing but equation $$(1)$$ as we derived.

• Okay. I’m just making notes. You said “The major point is the reaction starts in acidic medium and stay acidic at the end.” Maybe but if we reduce the pH by 8 moles of H+, aren’t we at least changing it? It seems that by adding these ions I would be changing the pH.
– BSD
Apr 5 '20 at 21:29
• “Now we have extra hydrogen in product side, so we don't have a choice but balance that by H+ in reactant side“ okay, that makes sense. Then you add OH on the reactant side because the solution is neutral. I can see that. The 4H+ had to come from somewhere leaving the OH- behind.
– BSD
Apr 5 '20 at 21:32
• Alright I’ve read it all closely, but I still don’t understand why the pH should change. “The major point is this reaction starts in neutral medium, but become basic at the end.” If I add these ions, I wouldn’t expect any of them to alter the pH. I is not a base. I suppose MnO4- is a conjugate base, but I would expect to see the extra H+ in that case. It does not make sense to me for anything but a base to make the solution more basic. Please explain.
– BSD
Apr 5 '20 at 21:39
• Comment 1: Remember, adding $\ce{H+}$ to equation doesn't mean you are adding to solution. Solution is acidic already by the using conditions, we are just using the exsisting condition to balance the equation. Apr 5 '20 at 22:59
• This is true for the remaining comments. For last one, $\mathrm{pH}$ increases as reaction progress. Apr 5 '20 at 23:02

If the reaction happens at neutral or mildly alkalic $$\mathrm{pH}$$:

$$\ce{6I- + 4 H2O + 2 MnO4- -> 3 I2 + 2 MnO2 (s) + 8 OH-}$$

If the reaction happens at mildly aciding $$\mathrm{pH}$$, it could be as well:

$$\ce{6I- + 8 H+ + 2 MnO4- -> 3 I2 + 2 MnO2 (s) + 4 H2O }$$

In any case, either spending $$\ce{H+}$$ either production $$\ce{OH-}$$ increases $$\mathrm{pH}$$. Both iodide and permanganate carry a negative charge while their spent forms are neutral. This negative charge must be passed to other anions like $$\ce{OH-}$$ or neutralized by $$\ce{H+}$$.

If one considers further disproportionation of iodine to iodide and iodate in alkalic solutions, then the total reaction is:

$$\ce{ I- + H2O + 2 MnO4- -> IO3- + 2 MnO2(s) + 2 OH-}$$

In strongly alkalic it could be also $$\ce{ I- + 6 OH- + 6 MnO4- -> IO3- + 6 MnO4^2- + 3 H2O }$$

It may be obvious if the equation is written together with spectator ions:

Mildly acidic: $$\ce{6KI + 8 HA + 2 KMnO4 -> 3 I2 + 2 MnO2 (s) + 4 H2O + 8 KA}$$ resp neutral/mildly alkalic $$\ce{6KI + 4 H2O + 2 KMnO4 -> 3 I2 + 2 MnO2 (s) + 8 KOH}$$ resp alkalic $$\ce{ KI + H2O + 2 KMnO4 -> KIO3 + 2 MnO2(s) + 2 KOH}$$ resp. strongly alkalic $$\ce{ KI + 6 KOH + 6 KMnO4 -> KIO3 + 6 K2MnO4 + 3 H2O }$$

• Okay. That does help, but KOH is still a base and would still affect the pH. In other words, I add KI and KMnO4 which are both neutral and I suddenly arrive at a base. Why?
– BSD
Apr 5 '20 at 21:43
• Thanks for editing the notation. I’m not familiar with it but I’ll try to figure it out.
– BSD
Apr 5 '20 at 21:44
• Because all anions of acids are removed and nothing but OH- can carry negative charge. Do you see other option ? If suddenly OH- is created, pH suddenly increases. Apr 6 '20 at 5:53

For the reaction:

$$\ce{6I- (aq) + 4 H2O + 2 MnO4- (aq) ->[excess I-] 3 I3- + 2 MnO2 (s) + 8OH-}$$

what you are missing is the Nernst Equation. The two reduction half cell reactions are:

$$\ce{I3− + 2e- <=> 3I− \quad\quad EMF = +0.53}$$

$$\ce{MnO4− + 2H2O + 3e− <=> MnO2(s) + 4 OH− \quad\quad EMF =+0.595}$$

The gist of the Nernst Equation is that the voltages change as the concentration of the ionic species change. The standard EMF is for 1 molar concentration of all ionic species.

For the $$\ce{I3-/I-}$$ half cell there is no pH consideration in that HI is a strong acid.

For the permanganate reduction:

• As the permanganate is reduced and the hydroxide builds up, the Nernst equation shows that the EMF of the half cell drops towards zero.

• If the initial permanganate was 1 molar and the hydroxide was 0.1 molar, then the EMF would be even more positive than 0.595 V.

As a bit of a side note $$\ce{I2}$$ has a very limited solubility in solution, hence an excess of $$\ce{I-}$$ is typically used so that the soluble $$\ce{I3-}$$ complex is formed.


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# Information Security and Cryptography Research Group

## A Provably-Secure Strongly-Randomized Cipher

### Ueli Maurer

Advances in Cryptology — EUROCRYPT '90, Lecture Notes in Computer Science, Springer-Verlag, vol. 473, pp. 361–373, May 1990, Final version: [Maurer92b].

Shannon's pessimistic theorem, which states that a cipher can be perfect only when the entropy of the secret key is at least as great as that of the plaintext, is relativized by the demonstration of a randomized cipher in which the secret key is short but the plaintext can be very long. This cipher is shown to be “perfect with high probability”. More precisely, the eavesdropper is unable to obtain any information about the plaintext when a certain security event occurs, and the probability of this event is shown to be arbitrarily close to one unless the eavesdropper performs an infeasible computation. This cipher exploits the assumed existence of a publicly-accessible string of random bits whose length is much greater than that of all the plaintext to be encrypted; this is a feature that our cipher has in common with the previously considered “book ciphers”. Two modifications of this cipher are discussed that may lead to practical provably-secure ciphers based on either of two assumptions that appear to be novel in cryptography, viz., the (sole) assumption that the enemy's memory capacity (but not his computing power) is restricted and the assumption that an explicit function is, in a specified sense, controllably-difficult to compute, but not necessarily one-way.

## BibTeX Citation

@inproceedings{Maurer90a,
author       = {Ueli Maurer},
title        = {A Provably-Secure Strongly-Randomized Cipher},
booktitle    = {Advances in Cryptology --- EUROCRYPT~'90},
pages        = 361--373,
series       = {Lecture Notes in Computer Science},
volume       = 473,
year         = 1990,
month        = 5,
note         = {Final version: \cite{Maurer92b}},
publisher    = {Springer-Verlag},
}


text:
Once you have completed instructions found in Installer Setup, you can then use the Rakefile contained at the top level of this repository (Rakefile). First you will run a rake task for copying measures and resource files from the OpenStudio-BEopt) repository into the top-level resources/measures folder. Then you will run rake task(s) for performing integrity checks on project inputs.

## Using the Rakefile¶

Run rake -T to see the list of possible rake tasks. The -T is replaced with the chosen task.

$rake -T rake copy_beopt_files # Copy files from Openstudio-BEopt ... rake download_and_copy_beopt_files # Download and copy files from Open... rake integrity_check_all # Perform integrity check on inputs... rake integrity_check_resstock_national # Perform integrity check on inputs... rake integrity_check_resstock_testing # Perform integrity check on inputs... rake test:all # Run tests for all rake test:regenerate_osms # regenerate SimulationOutputReport...  ## Copying Residential Files¶ To copy a set of residential measures and resource files to this repository’s resources/measures folder, run rake download_and_copy_beopt_files. You will be prompted to supply the branch name containing the set of residential measures of interest. $ rake download_and_copy_beopt_files
Enter branch of repo(<ENTER> for master):
<branch_name>

If you get any SSL or otherwise -related errors preventing your branch download, use an Internet browser to manually download the desired branch zip file (https://codeload.github.com/NREL/OpenStudio-BEopt/zip/<branch_name>). Move the downloaded zip file into the top level of this repository. Then run rake copy_beopt_files. You will not be prompted to supply a branch name.

Be sure to delete the zip file once all files have been extracted and copied.

## Integrity Checks¶

Run rake integrity_check_resstock_<project_name>, where <project_name> matches the project you are working with. If no rake task exists for the project you are working with, extend the list of integrity check rake tasks to accommodate your project by copy-pasting and renaming the integrity_check_resstock_national rake task found in the Rakefile. An example for running a project’s integrity checks is given below:

\$ rake integrity_check_resstock_national
Checking for issues with project_resstock_national/Location Region...
Checking for issues with project_resstock_national/Location EPW...
Checking for issues with project_resstock_national/Vintage...
Checking for issues with project_resstock_national/Heating Fuel...
Checking for issues with project_resstock_national/Usage Level...
...

If the integrity check for a given project fails, you will need to update either your tsv files and/or the resources/options_lookup.tsv file. See Options Lookup for information about the options_lookup.tsv file.


text:
# Show that limit of $\frac{\sin(xy)}{\sqrt{x^2+y^2}} = 0$ as $(x,y) \to (0,0)$

I'm asked to show that the limit of $\frac{\sin(xy)}{\sqrt{x^2+y^2}} = 0$ as $(x,y) \to (0,0)$. I guess the easiest way to do this would be by converting to polar form, so this is what I did:

$$\frac{\sin(xy)}{\sqrt{x^2+y^2}} = \frac{\sin(r^2\cos\theta \sin\theta)}{\sqrt{r^2\cos^2\theta+r^2\sin^2\theta}} = \frac{\sin(r^2\cos\theta \sin\theta)}{\sqrt{r^2(\cos^2\theta+\sin^2\theta)}} = \frac{\sin(r^2\cos\theta \sin\theta)}{r}$$

from here I guess I should try to prove using the definition, so I did the following:

since we know L is $0$

$$\left|\frac{\sin(r^2\cos\theta \sin\theta)}{r} - 0\right| = \left|\frac{\sin(r^2\cos\theta \sin\theta)}{r}\right| \leq \left| \frac{1}{r} \right| = \frac{1}{r}$$

which is true because $sin$ is bounded by $0$ and $1$, and $r$ is always a positive value. I also know that, since we are using polar coordinates, the distance between $(0,0)$ and $(x,y)$ is $r$. But I'm stuck here, I don't really know where to go from this. I guess I'm not getting the strategy behind the definition?

I think I'm pretty close to the answer, so feel free to give me the last steps if there's little to be done, it won't spoil the fun for me :)

• Use $|\sin\phi|<|\phi|$ as $\phi\to0$ – Nosrati Jan 30 '17 at 4:52

Hint: $\;\frac{\sin(xy)}{\sqrt{x^2+y^2}} = \frac{\sin(xy)}{xy} \cdot \frac{xy}{\sqrt{x^2+y^2}}\,$, then note that $\;\left|\frac{xy}{\sqrt{x^2+y^2}}\right| \le \sqrt{\frac{1}{2}|xy|}\;$ by AM-GM.
• The constant on the right hand side should be $1/\sqrt{2}$ (not $1/2$ as stated). – Fabian Jan 30 '17 at 5:38
$$0\le|\frac{\sin(xy)}{\sqrt{x^2+y^2}}|=\frac{|\sin[r^2\sin(\theta)\cos(\theta)]|}{r}\le \frac{|r^2\sin(\theta)\cos(\theta)|}{r}\le r\to 0$$


text:
# Difference between revisions of "2006 AIME II Problems/Problem 8"

## Problem

There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color.

Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed?

$[asy] pair A,B; A=(0,0); B=(2,0); pair C=rotate(60,A)*B; pair D, E, F; D = (1,0); E=rotate(60,A)*D; F=rotate(60,C)*E; draw(C--A--B--cycle); draw(D--E--F--cycle); [/asy]$

## Solution 1

If two of our big equilateral triangles have the same color for their center triangle and the same multiset of colors for their outer three triangles, we can carry one onto the other by a combination of rotation and reflection. Thus, to make two triangles distinct, they must differ either in their center triangle or in the collection of colors which make up their outer three triangles.

There are 6 possible colors for the center triangle.

• There are ${6\choose3} = 20$ possible choices for the three outer triangles, if all three have different colors.
• There are $6\cdot 5 = 30$ (or $2 {6\choose2}$) possible choices for the three outer triangles, if two are one color and the third is a different color.
• There are ${6\choose1} = 6$ possible choices for the three outer triangles, if all three are the same color.

Thus, in total we have $6\cdot(20 + 30 + 6) = \boxed{336}$ total possibilities.

## Solution 2 (Easy burnsides)

We apply Burnside's Lemma and consider 3 rotations of 120 degrees, 240 degrees, and 0 degrees. We also consider three reflections from the three lines of symmetry in the triangle. Thus, we have to divide by $3+3=6$ for our final count.

Case 1: 0 degree rotation. This is known as the identity rotation, and there are $6^4=1296$ choices because we don't have any restrictions.

Case 2: 120 degree rotation. Note that the three "outer" sides of the triangle have to be the same during this, and the middle one can be anything. We have $6*6=36$ choices from this.

Case 3: 240 degree rotation. Similar to the 120 degree rotation, each must be the same except for the middle. We have $6*6=36$ choices from this.

Case 4: symmetry about lines. We multiply by 3 for these because the amount of colorings fixed under symmetry are the exact same each time. Two triangles do not change under this, and they must be the same. The other two triangles (1 middle and 1 outer) can be anything because they stay the same during the reflection. We have $6*6*6=216$ ways for one symmetry. There are 3 symmetries, so there are $216*3=648$ combinations in all.

Now, we add our cases up: $1296+36+36+648=2016$. We have to divide by 6, so $2016/6=\boxed{336}$ distinguishable ways to color the triangle.

## Solution 3 (stars and bars)

There are $6$ choices for the center triangle. Note that given any $3$ colors, there is a unique way to assign them to the corner triangles. We have $6$ different colors to choose from, so the number of ways to color the corner triangles is the same as the number of ways to arrange $6 - 1 = 5$ dividers and $3$ identical items. Therefore, our answer is $$6 \binom{5 + 3}{3} = 6\binom{8}{3} = \boxed{336}.$$

-MP8148

Explanation of the bijection by WIlliamgolly: Let 1,2,3,4,5,6 be the colors, and WLOG assume that the middle triangle has a color of 6. Now, the color bijection can be formed as follows: Pick the colors to the immediate right of a divider. If there is no color to the immediate right of a divider, then that color is 6.

For example, ||12345| would represent the colors 1,1,6 as the colors chosen. Note for any three colors, there is only one way to fix it on the triangle, thus forming our stars and bars bijection.


text:
video studio lighting, what to buy

ccfan213

Active Member
my dad is a public relations officer for the army, and he preposed to his boss to turn a meeting room into a video studio. basically, the purpose of this room is to shoot video that will be edited and sent out to approx 3800 employees and soldiers. my dad's boss is a general and these videos will be highly circulated so it has to look good, but it also has to be low budget. My dad wanted to buy a simple set of track lights from home depot, but that didnt seem to me to be a wise choice. As a sound guy, i didnt have any information for him on what to buy, but i thought i would ask around to see if any light people had any ideas. Thank you for what im sure will be extremely helpful information

zac850

Well-Known Member
In general movie cameras--all cameras like a lot of white light on the subject. If you could find some scoops, or maybe even some old par's they would work. You really want to be looking for something that will give you a large, white, soft beam.

When lighting it, do not try to do anything 'artsy' or fantsy'. If you want to gel the lights, don't make it anything to dark, really don't go much darker then an R02.

ccfan213

Active Member
thanks zac, i really dont know much about lights, so getting fancy isnt a problem. but it reminded me of another criteria i left out, they must also be easy to operate, because there are no techs in my dads office, and they need to be able to turn this on.

The_Guest

Senior Team Emeritus
If there isn't drywall already installed, if possible setup a corner with dry wall and paint it solid green. It's helpful for editing and is nice for lighting. If not you can always by a solid green fabric. Also install metal bars from the cealing to hang lights, similarly to the pipes and bars you hang fixtures from in theaters. Try to get a bunch of power outlets wired on the ceiling, and if you have the cash install a cheap DJ dimmer system or just by a pack and clamp it to the bars. Run a the DMX or whatever interface up to there and get a simple manual board no larger than like 12 channels. Make sure it's all edison just to keep everything simple. As far as fixtures, just find cheap, used, and dj pars. Some of those cheap dj fixtures found on Musicians Friend what work good. Don't spend too much on anything because cameras need really basic lighting, just keep it all simple. Don't spend too much on a lighting system because what he really has to worry about is all the mics, cameras, editing, and all the other video equipment. That's where all the money is needed most. Good Luck.

Mayhem

Senior Team Emeritus
Are you able to purchase what we refer to as 4 pak dimmers in the US? Probably the most famous is the Jands 4 pak http://www.jands.com.au/jandsweb/lighting_dim_4pakII.html (of which I think all others are based upon). Essentially it is a 4 channel dimmer that has the dimming and control circuitry built in. You plug your generic lights directly into the back (using extension leads etc) - Nice and simple to use.

I would try to get one of these and several QI’s Pars or Fresnels. See if you can borrow for an initial set up. I have only used video on a handful of occasions and found that even lighting is better than directional lighting.

Also, seeing a lot of professional videographers at various gigs, I have noticed that most point a flood up at the ceiling to give an even coverage of light over the field.

Hope that this is somewhat helpful.

ccfan213

Active Member
thanks alot, this information will be very helpful, im sure my dad and his coworkers will be very appreciative so KEEP POSTING!

ship

Senior Team Emeritus
Shooting a video in a room can be complex but also dependant upon the cameras. The Blue screen/or green screen for instance cited would be a background used to isolate the person in front of it for adding a differing and changing background into it by way of editing such as your weatherman’s blue screen he does his forecast in front of. Were it just someone at a desk or podium in reading messages, such a background could be more if used at all instead of the company/division banner behind, more like used by the normal news people with the picture sized blue screen behind them with the logo or photo of what they were talking about. This if not instead with current technology, no need for a blue screen, the image is placed either on video monitors behind or even in front of the news person. Blue screens other than for weather forecasts are for the most part 1970s technology.

In any case, I’m sure your dad should probably be contacting someone at AFVN if I remember right as the military TV network for some expert advice much less perhaps gear for equipping him or the proper sources to shop at. Depending upon the cameras used and the need for light, it can be just some track lighting as a possibility, to some specialized KenoFlo flurorscent banks on C-Stands with incandescent type sources just added for toning.

We use some Juno track wash lights with 250w bulbs for instance in close proximity along with the Altman 650Q fixtures and Q-Lites for video shoots. MTV, Wanna be a DJ for instance was lit with these hung off a truss in a normal sized room. This in additon to Lekos and PARs with the high output lamps to add highlight and shadow over the general wash. A four Fresnel Arri Flex lighting kit having two three inch three hundred watt and two four and a half inch six fifty watt Fresnels plus various screens and filters is used otherwise with some wash lights during the NBA Playoff games for player interviews. Cinegel by Rosco would also be a useful part of the Rosco website to check into for technotes on using gel on film.

A question of color temperature and output most likely as to what fixtures you can use. A standard stage Fresnel has a fairly in-effective and dark lamp to be using. A high output lamp is not only at the cost of wattage have a high output but frequently also have a high color temperature that will read well on film. Remember also that shadows and what your eyes blend won’t be blended by the camera. For this reason illumination and washing out is frequently more a cover it all and add effect over this than lighting angles etc. in usefulness. Also you need more wash lighting so there is no drop off’s in intensity across the set as you move around it.

Dimming and control probably won’t be wise either. Remember that as you dim the lamps your color temperature and intensity drop with a spike towards the cooler amber shift. This amber shift on camera is much more drastic than with the eye. Instead of doing a lot of dimming with more than just the key lights as it were at most, but not the wash ones and in starting simple, I would think it would be much better to lamp the fixtures with the desired lamps to achieve intensity at their full output than to get something larger than need to dim them. Most studio lights have switches on them and it’s not only to turn off the draw before un-plugging them.

Here is some sources amongst many that might have more to read up on for what you need or want to do:
Bardwell & McAllister Inc. (Motion Picture Fixture & Rig Equipment Manufacturer/Dist.) http://www.bmllighting.com
Kino Flo (Fixture Mfr./Distributer) http://www.kinoflo.com/
Lowell Lighting (Omni Movie Lights) http://www.lowel.com/
MSE Studio Equipment, Inc/Matthews Studio Equipment (Movie Rigging Supplier) www.msegrip.com
FAZETRON STTV Lighting TV/AV/Video Sales & Service (Production Co/Dist.) http://www.stagelighting.com/
Studio Depot /Mole-Richardson (Theatrical Supply & Fixture Mfr..) Www.studiodepot.com

soundman

Well-Known Member
For our school morning broadcast they use 6 or 8 pars that just plug into a surge protector and that plugs into the wall so all they do before filming is flip a few switchs and roll camera. The pars are on a pipe suspened across two trees. Looks good too except one dark spot but its not on anyones face so they dont worry about it. For a set we just bult them a 3 person news desk with a flat behind them that has the schools logo on it. every thing was father dusted to give it a look.

ccfan213

Active Member
wow ship, that is extremely helpful, i dont even think it occured to him to contact the military TV network. thanks alot evereone and keep posting these helpful suggestions!

ship

Senior Team Emeritus
I used to watch a lot of AFVN or what ever it was called for a network. Back than it was the Liberal news media against the military thus the AFVN gave me balance in information given out. Now it's kind of the reverse more frequently. Dennis Miller gone Republican, what's the world coming to?

Been about 10 years now since it was available on my cable network since I moved. Decent network for information after the propaganda.

In any case, I hope they will be helpful. There is also lots of books out there on video and TV lighting that could be helpful.

dvsDave

Benevolent Dictator
2 Bogen swivel umbrella adapter .......$26.95 @$53.90


text:
# Comparison between geologic maps of Ethiopia

### Detailed Description

Comparison between geologic maps of Ethiopia, the first drawn in 1972 over a topographic map from 1968 and the second consisting of a 2004 Landsat image georeferenced on SRTM. Courtesy of Radar Technologies International.


text:
# Thread: MATH2901 Higher Theory of Statistics

1. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by leehuan
Oh I double checked that. I put the n back in there, got a determinant of (2alpha beta^2)/n^2 * (a matrix with n's appearing everywhere). That's fine I reckon

When we were taught it, we were honestly taught "educated guess". Normally I try to exploit the CLT.
Oh yeah, I forgot to include the n in front of the matrix when getting the 1,1 entry of the inverse. It should be n in the denominator indeed (for final answer).

2. ## Re: MATH2901 Higher Theory of Statistics

Can you guys help me with going through this question:

So to start off, is the null and alternative hypothesis.... H0 = 95 versus H1 < 95. Or is this wrong??

OR is it H0 < 95 versus H1 >= 95

nvm

4. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by Flop21
Also I read that if standard deviation / sigma is know, we can use z-distribution. If not, use t-distribution. We're given the sample SD right?? Does that mean we know it or not?
We don't know it ("it" refers to the population standard deviation (or really the standard deviation of the assumed underlying distribution)), since that's only the sample standard deviation.

5. ## Re: MATH2901 Higher Theory of Statistics

Help pls someone,

So we have to use R commands instead of distribution tables (wtf it's so much easier with the tables)?

How would you then find the region of rejection (when doing a hypothesis test) for both z and t distributions?

E.g. for T test, I would find the degrees of freedom (n-1), and look up n-1 on the T table with the respective value of alpha, say 0.05. How would you do this in the test????

6. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by Flop21
Help pls someone,

So we have to use R commands instead of distribution tables (wtf it's so much easier with the tables)?

How would you then find the region of rejection (when doing a hypothesis test) for both z and t distributions?

E.g. for T test, I would find the degrees of freedom (n-1), and look up n-1 on the T table with the respective value of alpha, say 0.05. How would you do this in the test????
You leave it there. They will supply you the R code and all you have to do is read it.

Since you do computer science you shouldn't have any trouble understanding just one line of code tbh

7. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by leehuan
You leave it there. They will supply you the R code and all you have to do is read it.

Since you do computer science you shouldn't have any trouble understanding just one line of code tbh
yeah I am a mind reader that knows what vaguely named functions do

But anyway I know it's just going to be a shit show. Hopefully not, but I am 80% sure it will be.

8. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by Flop21
yeah I am a mind reader that knows what vaguely named functions do

But anyway I know it's just going to be a shit show. Hopefully not, but I am 80% sure it will be.
R's free to download. You can always just investigate briefly :P

Here's an example of something that might appear

Tbh all I've seen is p___, q___, r___ where ___ is a distribution.
p - probability
q - quantile
r - just a bunch of random simulations

9. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by leehuan
R's free to download. You can always just investigate briefly :P

Here's an example of something that might appear

Tbh all I've seen is p___, q___, r___ where ___ is a distribution.
p - probability
q - quantile
r - just a bunch of random simulations
Thanks

tbh I think I will fail this course unfortunately

10. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by Flop21
Thanks

tbh I think I will fail this course unfortunately
You could try the 2901 exam past papers just in case, as they seem to be similar difficulties (I think some questions overlap? or the old ones used to anyway.)

11. ## Re: MATH2901 Higher Theory of Statistics

All this stat stuff looks so much more complicated than what I've done. Only done stat171 where had z, various t tests, and chi squared test for independence and GOF test... rip

12. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by BenHowe
All this stat stuff looks so much more complicated than what I've done. Only done stat171 where had z, various t tests, and chi squared test for independence and GOF test... rip
We learn this in second year at UNSW, so it seems a bit later than at Macquarie.

13. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by Flop21
Can you guys help me with going through this question:

So to start off, is the null and alternative hypothesis.... H0 = 95 versus H1 < 95. Or is this wrong??

OR is it H0 < 95 versus H1 >= 95
I think youre supposed to make the null hypothesis equal to something but it doestn really matter so I guess $H_{0}:\ C\geq\ 95, H_{a}:\ C<95.\text{where C is the cholesterol level}\\ t_{obs}=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}\\ t_{obs}=\frac{89.02-95}{\frac{12.9}{\sqrt{6}}}=-1.14\\ \text{Since this is a 1 tail test (note left side), p value}=P(t<-1.14)\\ \text{From the t table and noting DF equals 5, p value}>0.1\\ \text{Assuming alpha is 0.05, at the 5 percent significance level there is insufficient evidence to reject}H_{0}\\ \text{i.e. The blood cholesterol calcium content is greater than or equal to 95 mg/dl for the rat population with 5 percent added fibre to their diets}$

This is how I would do it

14. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by BenHowe
I think youre supposed to make the null hypothesis equal to something but it doestn really matter so I guess $H_{0}:\ C\geq\ 95, H_{a}:\ C<95.\text{where C is the cholesterol level}\\ t_{obs}=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}\\ t_{obs}=\frac{89.02-95}{\frac{12.9}{\sqrt{6}}}=-1.14\\ \text{Since this is a 1 tail test (note left side), p value}=P(t<-1.14)\\ \text{From the t table and noting DF equals 5, p value}>0.1\\ \text{Assuming alpha is 0.05, at the 5 percent significance level there is insufficient evidence to reject}H_{0}\\ \text{i.e. The blood cholesterol calcium content is greater than or equal to 95 mg/dl for the rat population with 5 percent added fibre to their diets}$

This is how I would do it
I think it should be Ho: U = 95, Ha: U < 95

R output is mainly qnorm / pnorm / qt IIRC. Qnorm is % shaded eg qnorm(0.5) = 0 i.e. 50% shaded is at 0 on the normal distribution, etc. Pnorm is opposite so Pnorm(0.5) = the % shaded at z = 0.5 (or the pnorm/qnorm may have been other way around)

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