Dynamics of Blade With Geometrical Nonlinearity
of Nonstationary Vibrations, A.N. Podgorny Institute for Mechanical Engineering
The compressor blade is described by pre-twisted, variable thickness, double curvature shallow shell with geometrical nonlinearity. The R-function method and the Rayleigh- Ritz approach are used collectively to calculate eigenmodes of linear vibrations. Nonlinear vibrations of shell are approximated by using these eigenmodes. The obtained finite degrees-of-freedom system has three internal resonances. The free nonlinear vibrations are studied by nonlinear modes.
Keywords: shallow shells, finite degree-of-freedom system, internal resonances, nonlinear normal modes
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Significant dynamics loads act on the blades of turbo machine during their operation. Therefore, blades can perform vibrations with amplitudes commensurable with their thickness. Such blade deformations are described by geometrical nonlinear theory. Many efforts have been done to study blade vibrations. Venkatsan, Nagaraj (1982) studied nonlinear vibrations of rotating blades. They came to the conclusions, that the frequency response can be hard or soft. The vibrations of turbine blades under the action of longitudinal periodic force have been considered by Chen, Peng (1995). Using geometrically nonlinear theory and finite element method, the blade nonlinear model is obtained. The vibrations of shallow anisotropic blades are treated by Abe et al. (2000). They used the Rayleigh-Ritz method to analyze linear vibrations. Nonlinear vibrations are expanded into two linear modes. The finite degrees-of- freedom model is obtained by the Galerkin procedure. Liew, Lim (1996) used energetic approach to study linear vibrations of shallow shells with different Gaussian curvature and rectangular base. Nonlinear vibrations of hydraulic turbine blades, which are modeled by pre-twisted shell with variable thickness and ring sector-shaped base, are treated by Hu, Tsuiji (1999), Sakiyama et al. (2002). The dependence of eigenfrequencies and eigenmodes on the pre-twisted angle and thickness are investigated. The papers of Hoa (1981) and Ross (1975) are devoted to linear vibrations analysis of blades, which are modeled by shells. The results of finite element analysis of turbo machine blades are compared with the experimental data of Mindle and Torvik (1987). Didkovskii (1974) analyzed the parametric vibrations of turbo machine blades in gas flow. The sufficient conditions of dynamical stability are obtained. Nabi and Ganesan (1996) compared the beam and plate models of turbo machine blades. They came to the conclusion that the plate models are better. Pisarenko, Vorob’ev (2000) concluded that the theory of pre-twisted beams gives the adequate eigenfrequencies of blade vibrations. However, the more accurate models should be used for detailed stresses analysis. Choi, Chou (2001) analyze the blade vibrations with account of shear. The influence of shroud on vibrations is considered. Pak et al. (1992) modeled the turbo machine blades by thin elastic beams. Flexural, torsion and flexural-torsion vibrations are analyzed in detail. Breslavsky et al. (2011) studied nonlinear multi-mode vibrations of hydraulic turbine blade, both in vacuo and immersed in the fluid. The blade is modeled by shallow shell with ring sector base.
nonlinear vibrations of compressor blades are treated in this paper.
2. The problem formulation
The vibrations of the compressor blades (Figure 1) are considered. The blades are modeled by pre-twisted shallow shell with trapezoidal base and variable thickness. The sketch of the shell base is presented in Figure 2. The shell is clamped on one side and it is free on others three sides. The coordinate plane () passes through the chord of the root section and the central line of the blade. Moreover, the axis coincides with the central line of the blade. The geometry of the shell middle surface is complex, therefore, the shell is not considered in principal coordinates.
The compressor blade previously studied by Meerovich (1956, 1961) is treated. The shape of the untwisted shell middle surface can be described by the following function:
where is the distance between the middle surface and the plane xy in the root section; is the length of the root section chord; ; is the distance between the middle surface and the plane () on the peripheral edge; is the length of the blade. The thickness of the untwisted blade can be presented in the following analytical form (Meerovich 1961, 1956):
where is the maximal thickness of the root section; is the maximal thickness of the peripheral edge.
It is assumed, that the blade is pre-twisted uniformly and pre-twisted angle is denoted by . Then the thickness and the shape of middle surface can be presented as:
The boundary conditions on the clamped edge are the following:
where , are displacements of the middle surface points in x, y, z directions, respectively.
As thin shell is considered, the shear is not taken into account. Then the potential energy of the shell can be presented as (Grigoluk, Kabanov, 1978; Goldenveizer et al. 1979):
where are the Young’s modulus and the Poisson’s ratio of the shell material. The parameters of shell deformations are determined in the following form (Grigoluk, Kabanov, 1978):
; ; ; (3)
; ; ;
; ; ,
where are linear components of membrane strains of shell middle surface; are components of the bending deformations of the middle surface; and are rotation angles about the tangent vectors to the lines x and y, respectively; is angle of rotation about normal to the middle surface. Curvatures of the middle surface are determined as:
; ; ,
where , are coefficients of the first and the second quadratic form of the middle surface.
The kinetic energy of the shell has the following form (Amabili, 2008):
where is the shell material density.
3. Linear vibrations analysis
In this section the linear vibrations of the shell are considered and nonlinear terms are not taken into account in the expressions (3). The Rayleigh-Ritz method is used to determine eigenfrequencies and eigenmodes of shell, so only the kinematic boundary conditions are taken into account. The boundary conditions on the shell free edges are natural. Therefore, they are not taken into account in linear analysis.
The periodic vibrations of the shell are presented in the following form:
R-function (Rvachev, Sheiko, 1995) is used to satisfy the boundary conditions (1).
R-functions are very useful for shallow shells with complex base analysis (
The function can be derived for arbitrary domains with analytical boundaries. The methods for these functions construction are considered by Rvachev, Sheiko (1995).
According to the R-function theory, the clamped part of the shell boundary can be presented by the following function:
where is the operation of R-conjunction. The functions and describe axis and the domain , respectively. These functions can be presented in the following form:
The steps of the function construction are shown in Figure 3.
Following the Rayleigh-Ritz method, the vibrations modes are presented as:
where , , are trial functions. The expressions (5) satisfy boundary conditions (1). The following trial functions are used in this paper:
where describes the trapezoidal base of the shallow shell.
The expressions (5) are substituted into the Lagrange functional and the integration is carried out. Then the functional is obtained as quadratic form of the parameters . In general, can be presented in the following form:
The minimum of the Lagrange functional is determined from the following equations: ;. These equations can be presented as the eigenvalue problem:
4. Nonlinear finite degrees-of-freedom model of vibrations
Nonlinear vibrations of the blade are approximated by the eigenmodes of linear vibrations. Then the shell bending vibrations can be presented as:
where are normalized eigenmodes. The displacements and are presented as:
where and are eigenmodes of in-plane vibrations. Now the equations (9, 10) are substituted into the kinetic and potential energy (2, 4) and the integration is carried out. As a result, the potential and kinetic energies are obtained as functions of the generalized coordinates and the generalized velocities in the following form:
Using the kinetic and potential energy, the Lagrange equations with respect to the generalized coordinates are derived.
Due to the pre-twist of the shell, the displacements orthogonal to the blade middle surface consist of both displacements and . The displacements have moderate amplitudes. Therefore, the inertial terms only in the direction are not taken into account. Then the system of ordinary differential Lagrange equations is transformed into the system of linear algebraic equations:
and the system of ordinary differential equations. The solutions of linear algebraic equations are substituted into the system of ordinary differential equations. The following dimensionless variables and parameters are used in future analysis:
, , , (11)
where is the first (minimal) eigenfrequency. Then the finite degrees-of-freedom dynamical system with respect to dimensionless generalized coordinates and parameters has the following form:
where are known system parameters. The matrix can be presented as
where is matrix of dimensionless eigenfrequencies. By means of the coordinate transformations
the system (12) is rewritten with respect to the modal coordinates:
5. Nonlinear normal modes of vibrations
Shaw-Pierre nonlinear normal modes are used to analyze the dynamical system
(13) (Shaw, Pierre, 1993; Jiang et al. 2005; Avramov, 2008;
Due to these internal resonances, the shell motions have three active generalized coordinates . Then the invariant manifold of the system can be presented as (Jiang et al. 2005):
This invariant manifold satisfies the following partial differential equations:
The functions can be presented as truncated power series as:
where , , are unknown coefficients. As the system (13) is presented with respect to the modal coordinates, the coefficients of summands (17) with degrees lower than two are equal to zero (Jiang et al. 2005). The solutions (17) are substituted into (16) and the summands with are equated. As a result, the set of the systems of linear algebraic equations is obtained. The first system is linear with respect to the coefficients of the second degrees (i.e. the summands with ) and the second system is linear with respect to the coefficients of the third degrees (i.e. summands with ). The coefficients of the series (17) are obtained by solving sequentially these systems of linear algebraic equations. Thus, the nonlinear mode (15) can be obtained.
The equations (17) are substituted into the third, fourth and fifth equations of the system (13). As a result, the system of three ordinary differential equations describing the motions on nonlinear modes is obtained in the following form:
, . (18)
The harmonic balance method is used to study the motions on the nonlinear mode (18). The solutions of the system (18) are presented as:
, . (19)
Substituting the expressions (19) into the equations (18), the system of nonlinear algebraic equations with respect to ; is derived. This system is solved by the continuation technique (Seydel, 1997).
To analyze stability of periodic motions, the system (13) is rewritten in the following vector form:
where is phase vector, is vector function of . The stability of periodic motion is analyzed. Following the book (Yakubovich, Starzhinskii, 1975), small perturbation is added to periodic motions. The evolution of small perturbations in time is described by the following system of ordinary differential equations:
where is the Jacobi matrix.
Following the book (Parker, Chua, 1989), the fundamental matrix of the system (21) is calculated to analyze stability numerically. This matrix is a solution of the system (21) with initial conditions in the form of the identity matrix . The fundamental matrix at (where is vibrations period) is called monodromy matrix (Yakubovich, Starzhinskii, 1975). The conclusions about the stability of periodic motions are made according to the eigenvalues of monodromy matrix.
6. Numerical results
In order to perform numerical analysis the steel compressor blade is considered with the following parameters (Meerovich, 1961): m, m, m, , m, , , N/m2, , kg/m3.
The calculations of linear vibrations are carried out by the Rayleigh- Ritz method. The results of the eigenfrequency calculations are presented in the first row of Table 1. The experimental and numerical data of the previous research, which are obtained by Meerovich (1961, 1956), are presented in the second and the third row of Table 1.
Figure 4 shows the nodal lines of the first six eigenmodes of the blade vibrations. These results are close to the data from (Meerovich, 1961, 1956). However, the eigenfrequency, which is close to the fourth one, is leaved out in (Meerovich, 1961).
first five eigenmodes are used in the expansions (9, 10) to obtain the finite
degrees-of-freedom model of the blade nonlinear vibrations. Excluding the
generalized coordinates , the system with ten degrees-of-freedom in the form of (13)
is derived. The
The vibrations with dominant generalized coordinate , which is excited due to the internal resonance (14), are analyzed. In this case, the generalized coordinate is close to zero. The backbone curve of these motions is presented in Figure 5, which shows the vibrations amplitudes versus . The stable and unstable motions are shown on this figure by solid and dotted lines, respectively. The eigenfrequencies of linear vibrations are shown by vertical lines in Figure 5. The free vibrations (Figure 5) are stable until the amplitude of is small. If the frequency is increased, the vibrations amplitude decrease quickly. Note that the vibrations and have the dominant harmonics with the frequency . As follows from Figure 5, the free nonlinear vibrations become unstable at certain values of the vibration amplitudes. The frequencies of nonlinear vibrations (Figure 5) are not close to the frequencies of linear ones. Therefore, the geometrical nonlinearity affects essentially on the shell vibrations.
the vibrations with dominant generalized coordinate are analyzed. Such
motions are denoted by the number
Now another type of motions on the nonlinear mode (15) is considered. In this case the vibrations have significant amplitudes; the rest generalized coordinates are not excited. These vibrations are shown in Figure 7. The free nonlinear vibrations become unstable at some value of vibrations amplitudes.
The harmonic balance method is used directly to the system of nonlinear ordinary differential equations (13) to obtain the motions, which differ from the vibrations on the considered nonlinear normal mode. As a result, additional type of periodic motions is obtained. The generalized coordinate in this regime is active too. These vibrations are shown by the branch 2 on Figure 6. This branch describes the periodic motions, which split off the motions 1 due to the period doubling bifurcation.
In order to verify the obtained results the direct numerical integration of the system (13) is carried out. The initial conditions are obtained from the solution (19). The results are shown in Figure 5 and Figure 7 by dots. The data of the direct numerical integration are very close to the results of the normal mode analysis. So, the method suggested in the previous section adequately describes the system vibrations.
Now the projections of the middle surface points vibrations on the normal to this surface are treated. The corner points with the coordinates and have maximal displacements. However, the shell in these points is very thin. Therefore, the displacements of the shell points at small distance from the corner points are considered. Due to pre-twist of the blade, the displacements in the orthogonal direction to the middle surface contain both and . These displacements can be presented as
The backbone curves are shown in Figure 8, where the vibrations amplitudes versus the frequency are plotted. Figure 8 shows the backbone curves of the different vibration modes. The horizontal line shows the thickness of the shell in the point, where the vibrations are determined. The motions with dominant generalized coordinate and with dominant coordinate are shown on Figure 8a and Figure 8b, respectively.
The compressor blade is described by cantilever, pre-twisted, double curved, variable thickness shallow shell with trapezoidal base. At the first phase of the analysis the eigenfrequencies and eigenmodes are determined. The three internal resonances between the third, the fourth and the fifth vibrations eigenfrequencies are observed in the system. The different modes of the compressor blade nonlinear vibrations are described by hard backbone curves. It is shown that the amplitudes of stable vibrations of peripheral edge are commeasurable with the blade thickness in this region.
The free nonlinear vibrations become unstable at certain values of the vibrations amplitudes. Moreover, the frequencies of nonlinear vibrations are not close to the frequencies of linear ones. Therefore, geometrical nonlinearity affects essentially on the shell vibrations.
Abe, A., Kobayashi, Y., and Yamada, G., 2000. Non-linear vibration characteristics of clamped laminated shallow shells. J. Sound Vib. 234, 405-426.
Amabili, M., 2008. Nonlinear vibrations and stability of shells and plates.
Avramov, K.V., 2008. Analysis of forced vibrations by nonlinear modes. Nonlinear Dyn. 53, 117-127.
Avramov, K.V., Tyshkovets, O., Maksymenko-Sheyko, K.V., 2010. Nonlinear dynamics of circular plates with cutouts. R-function method. ASME J. Vib. Acoust. 132, ¹5, 110-135.
Breslavsky, I.D., Strel’nikova, E.A., Avramov, K.V., 2011. Dynamics of shallow shells with geometrical nonlinearity interacting with fluid. Comput. Struct. 89, 496-506.
Chen, L.W., Peng, W.K., 1995. Dynamic stability of rotating blades with geometrical non-linearity. J. Sound Vib. 187, 421-433.
Choi, S.-T., Chou, Y.-T., 2001. Vibration analysis of elastically supported turbomachinery blades by the modified differential quadrature method. J. Sound Vib. 240, 937-953.
Didkovskii, V.N., 1974. Parametric vibrations of turbomachines in a flow. Strength Mater. 8, 9-13.
Goldenveizer, A.L., Lidskii, V.B., Tovstik, P.E.,
1979. Free vibrations of thin elastic shells. Nauka,
Grigoluk, E.I., Kabanov,
V.V., 1978. Stability of shells. Nauka,
Hoa, S.V., 1981. Vibration frequency of a curved blade with weighted edge. J. Sound Vib. 79, 107-119.
Hu, X.X., Tsuiji, T., 1999. Free vibration analysis of curved and twisted cylindrical thin panels. J. Sound Vib. 219, 63-68.
Jiang, D., Pierre, C., Shaw, S.W., 2005. The construction of non-linear modes for systems with internal resonance. Int. J. Non-Linear Mech. 40, 729-746.
Liew, K.M., Lim, C.W., 1996. Vibration of doubly-curved shallow shells, Acta Mech. 114, 95-119.
Meerovich, I.I., 1956. Vibrations of slightly
curved and pre-twisted blades. Trudy
CIAM 271. Oborongiz,
Meerovich, I.I., 1961. Stress distribution in vibrating compressor blades. Oborongiz,
Mindle, W.L., Torvik, P.J., 1987. Multiple modes in the vibration of cantilevered shells. J. Sound Vib. 115, 289-301.
Mohamed Nabi, S., Ganesan, N., 1996. Comparison of beam and plate theories for free vibrations of metal matrix composite pre-twisted blades. J. Sound Vib. 189, 149-160.
Parker T.S., Chua L.O.,
1989. Practical numerical algorithms for chaotic systems.
Pisarenko, G.S., Vorob’ev, Yu.S., 2000. Issues of simulation of turbomachine blade vibration. Strength Mater. 32, 487-489.
Ross, C.T.F., 1975. Free vibration of thin shells. J. Sound Vib. 39, 337-344.
Rvachev, V.L., Sheiko, T.I., 1995. R-function in boundary value problem in mechanics. Appl. Mech. Rev. 48, 305-316.
Sakiyama, T., Hu, X.X., Matsuda, H., Morita, C., 2002. Vibration of twisted and curved cylindrical panels with variable thickness. J. Sound Vib. 254, 481-502.
Seydel, R., 1997. Nonlinear computation. Int. J. Bifurc. Chaos 7, 2105-2126.
Shaw, S.W., Pierre, C., 1993. Normal modes for nonlinear vibratory systems. J. Sound Vib. 164, 58-124.
Venkatsan, C., Nagaraj, V.T., 1982. Non-linear flapping vibrations of rotating blades. J. Sound Vib. 84, 549-556.
Yakubovich, V.A., Starzhinskii, V.M., 1975. Linear Differential Equations with Periodic Coefficients, Wiley,
Fig. 1. Sketch of the blade
Fig. 2. Sketch of the shell base
Fig.3. The steps of the construction of the function , which describes the clamped part of the shell boundary. The regions, where the function is positive, are shown by grey color. The black lines show the zero of the function.
a. b. c. d. e. f.
Fig.4. The nodal lines of eigenmodes corresponding to the following eigenfrequencies in Hz:
a. b. c. d.
Fig. 5. The backbone curves of the vibrations with dominant general coordinate
Fig. 6. The backbone curves of the free unstable nonlinear vibrations
Fig. 7. The vibrations of the shell with dominant general coordinate
Fig.8. The backbone curves of shell vibrations with dominant general coordinates: a. ; b. .
Table 1. Eigenfrequencies of vibrations
Eigenfrequencies in Hz
The results of the calculations
The results of previous calculations
The experimental data (Meerovich, 1961)


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Bayesian inference results in more- or equally-accurate diagnostic classification, especially with complex and noisy data sets such as those found when developing IVDs.
The statistical analysis methods that have primarily been taught in schools follow the frequentist approach. However, another approach, Bayesian statistics, is becoming more widespread. With today’s noisy data sets, Bayesian techniques allow scientists to extract the most information from an experimental data set, which can lead to the development of more-accurate diagnostic instruments.
FDA has recently recognized the value of Bayesian analysis compared with frequentist methods in conducting clinical trials for medical devices. Bayesian statistics provide a coherent way to learn from data as they accumulate and a formal mathematical technique for combining data with prior information. This can lead to smaller amounts of data needed in clinical trials for devices. The same argument holds in applying Bayesian techniques to the development of processing algorithms for diagnostic instruments. Devices suited for Bayesian techniques include microarrays, flow cytometry, proteomics, and time-of-flight spectroscopy. Such devices produce data that can be handled effectively by Bayesian analysis.
Frequentists and Bayesians stridently advocate their statistical approach, bordering on religious fervor. This article compares Bayesian and frequentist methods and illustrates applications in which the lesser-used Bayesian methods have significant advantages, in particular in developing diagnostic products. Because Bayesian methods ask and answer the right questions, they can lead to the development of more reliable, effective, and low-cost diagnostic solutions.
Description of Bayesian and Frequentist Methods
Bayesian techniques have become more widely used in real-world applications only during the last 30 years due to advances in the computing power required in the initial algorithm development and validation. However, once created, the algorithms can be embedded on low-cost digital signal processing chips. Although applying Bayesian inference to real-world situations requires an experienced user, the benefits that this technique offers far outweigh the initial investment.
Bayesian and frequentist methods differ in how the correspondence is constructed between mathematical objects and real-world ideas. The frequentist approach regards probabilities as measures of frequency, while the Bayesian approach regards them as degrees of belief.
In Bayesian statistics, what is known before collecting the data is the prior information, and what is known after collecting the data is the posterior information. In mathematical terms, the Bayesian technique calculates P(θ|D), the posterior probability distribution of the unknown variable θ given the data D (which is what the user actually wants to know but requires the prior probability distribution of θ). In comparison, the frequentist analysis calculates P(D ∈C|H0), the probability that the data fall in some critical region given some null hypothesis H0 about θ.
The difference between these two is pivotal. While the latter probability can be calculated without knowing the prior distribution of θ, the result is not the answer that a user actually needs. It is similar to answering a different question than what is asked in an exam because of a lack of knowledge on how to answer the original question.
Nonstatisticians often want to ask such questions as, “Does this set of samples come from a normally, also referred to as a Gaussianly, distributed variable?” In order to give meaning to such a question, a prior distribution on the distributions considered possible for the variable is essential. For example, if someone asks, “Is Martin 1.5 m tall?,” the answer depends on whether the word exactly is part of the question. If it is, then the answer is no with 100% probability. But if approximately is part of the question, then the answer depends on the exact meaning of approximately.
The same applies to questions about distributions. Very few experimentally observable variables are exactly Gaussianly distributed, even if they are approximately so. But in this case, the whole question turns on what exactly is meant by “approximately,” and without that, the question is meaningless. Unfortunately, specifying what is meant by approximately in this context is much more difficult than when measuring somebody’s height.
Figure 1. (click to enlarge) The prior distribution (left), which represents views on the probability of an ordinary coin coming down heads, and the posterior distribution given 7 out of 8 heads.
Meanwhile, the frequentist defines an unbiased estimator to be one such that the expectation of the estimator is always equal to the true value. Consequently, the frequentist maximum likelihood estimate, which is unbiased in the technical sense of the frequentist’s definition, comes in at 0.875. But this is unreasonable as an estimate of the probability of a coin coming down heads or tails, because it is biased by ignoring the prior information about coins. The combined effect of these factors is that users are not encouraged to think clearly about the problems they face. They get answers to the wrong questions, which are biased by ignoring information other than the data and concentrating on local areas of high probability density rather than the behavior of the whole probability mass.
The following are two more complex examples that illustrate the types of signal processing problems that could be encountered in a diagnostic instrument. These examples discuss how the choice between Bayesian and frequentist methods could influence diagnostic instrument designers. Depending on the application, using Bayesian inference may make the results more accurate, indicate precisely what uncertainties there are, or make the project possible at all rather than impossible. (There are many different frequentist methods. This article only mentions a subset because of space constraints.)
Fluorescence Lifetime Measurements
Extracting quantities of fluors of different lifetimes present in a mixture provides an example of how Bayesian techniques can generate better results than traditional frequentist methods. Fluorescence measurements are used in a variety of IVDs, from lateral-flow tests to high-throughput screening. This example will consider the situation of two fluors of known lifetimes present in unknown amounts.
The traditional method for solving this problem is time-correlated single- photon counting (TCSPC). A sample is illuminated with pulses of laser light that are deliberately so dim that only one fluorescent photon per hundred excitation pulses is expected to come back. This is an essential feature of this approach. Even if the laser could be made brighter, this approach would then fail: since only the first photon coming back after each excitation is recorded, the arrival times available would be biased to be earlier than expected.
After each excitation, the arrival time of only the first of any photons received is recorded. After many tens of thousands of excitation cycles, a histogram of the arrival times of the fluorescent photons seen is made, and least-squares curve fitting of a family of exponential curves is done. This method amounts to using maximum-likelihood estimation under the assumption that the errors are due to Gaussian noise of constant amplitude.
In contrast, the Bayesian method of solving this problem sets the laser to its full brightness (which we will assume is sufficient to get back on average one photon per excitation, but which will give even better results if a greater number of photons is received), records the arrival times not just of the first but of any and all photons received, and applies Bayesian inference to the resulting data.
In order to evaluate the two algorithms, the figures compare the Bayesian algorithm after n excitations and the traditional frequentist algorithm after 100n excitations, so that the average total number of photons emitted is the same for the two algorithms (although the collection time for the Bayesian algorithm is 100 times shorter). The Bayesian solution is shown by the background color plotting of the posterior, the correct result by an x, and the maximum likelihood point by a plus sign.
Figure 2. (click to enlarge) Results after 10 excitations for the Bayesian algorithm and 1000 excitations for the traditional method. The truth is shown by an X in a circle, the Bayesian answer is the colored area, while the traditional method’s answer is shown by a + in a circle.
Figure 3. (click to enlarge) Results after 10,000 excitations for the Bayesian algorithm and
1 million excitations for the traditional method. The truth is shown by an X in a circle, the Bayesian answer is the colored area, while the traditional method’s answer is shown by a + in a circle well to the left of the true point.
So in this example, the Bayesian method finds the true answer (by defining a range in which the truth lies) whereas the frequentist method gives the wrong answer and gives no indication that this is what it is doing. This has serious implications for any diagnostic device.
Despite having 100 times fewer excitations but the same overall average number of photons, several reasons explain why the Bayesian method produces better results than the traditional method. The reasons are that the Bayesian approach avoids the following problems:
- Biasing the photon arrival times by rejecting all but the first in any excitation cycle.
- Assuming Gaussianity inherent in least-squares curve fitting.
- Looking for the maximum of anything; rather, it returns information on the whole posterior probability mass.
Bayesian inference has enabled use of 100 times fewer excitations and nonetheless achieved a result that is genuinely unbiased compared with the traditional TCSPC method. The Bayesian method has also provided a precise view of the uncertainty remaining in a solution for any number of excitations, even as few as 10. Relating this to an diagnostic instrument, the Bayesian posterior distribution will encompass the correct results, while processing data with the frequentist method will lead to inaccurate results.
Automated Multivariate Diagnostics
Electronic noses illustrate the signal processing needed for nonspecific multivariate assay tests. An electronic nose is an array of nonspecific sensors of volatile compounds. While each sensor in the array can respond to a wide range of different compounds by giving an electrical signal, each sensor’s pattern of response to various compounds is different. The task for the signal processing block is to identify patterns of responses that can be used for diagnostic purposes.
A similar signal-processing philosophy can be applied to multigene tests that a number of different diagnostics companies are currently developing as cancer prognostics.1 This concept of an electronic nose will be used to compare the quality of results obtained with Bayesian and frequentist methods.
The developers of any statistical signal processing algorithm require an initial set of data, or training data, to determine the algorithm parameters. In this electronic nose example, we will suppose the nose distinguishes two groups: sick and well. In order to present and compare the results graphically, the number of sensors is limited to two (a value of 32 might be more typical).
Figure 4. (click to enlarge) The best possible classification certainty using information only available to the person who synthesized the data, and the 800 training data points.
Figure 5. (click to enlarge) The result of a Bayesian algorithm and the previously unseen points to be classified.
Figure 6. (click to enlarge) The result of applying maximum-likelihood model fitting.
Figure 7. (click to enlarge) The result of applying linear discriminant analysis (LDA).
The magnitude of the differences between the Bayesian solution and the other frequentist techniques can only be fully appreciated in more dimensions (i.e., with more component sensors in the electronic nose). This is analogous to analyzing a larger number of genes when looking for a gene signature as a method for diagnosing cancer.
Figure 8. (click to enlarge) Performance of Bayesian and various frequentist algorithms in 32 dimensions. Fraction of points wrongly classified of an unseen set of 200,000 points. MaxLike = maximum likelihood; LDA = linear discriminant analysis; NN = neural network; LR = logistic regression; PCA = principal components analysis.
Figure 8 also demonstrates how Bayesian inference has a 100-fold-lower error rate than other statistical analysis methods. In terms of a diagnostic instrument, implementing a Bayesian algorithm would better enable an electronic nose to make a correct diagnosis, or a gene test to better indicate better a predisposition to cancer.
Frequentist methods have been the predominant statistical analysis for the larger part of the last century, to the point where their adherents consider them to be the best way to analyze data and often defend them religiously. However, this article illustrates how Bayesian inference provides more accurate diagnostic classification, particularly with complex and noisy data sets such as those found when developing diagnostic products. Another benefit of Bayesian approaches is that this statistical technique sometimes allows cost savings to be made with the sensor hardware. For example, savings of 100-fold in memory usage and 10-fold savings in central processing unit cycles needed per second, along with a large improvement in performance, have been obtained.
In addition to automated multivariate diagnostics and fluorescence applications, Bayesian techniques can be applied to other diagnostic product categories that produce large-scale data, including microarrays, flow cytometry, proteomics, and time-of-flight spectroscopy.
Another important feature of Bayesian techniques is that the user is informed about the maximum amount of information that can be extracted from the available data. This means that when conducting the analysis, the user knows how good the results are. But with frequentist methods, the user is left wondering how much better the results could be. This advantage of Bayesian inference has not been discussed in this article due to space constraints but it is a significant advantage for device developers.
However, there are many situations in which Bayesian techniques are not needed to provide adequate performance, or where an approximation to Bayesian inference will provide adequate performance. Consequently, when optimal performance is not necessary, Bayesian inference may or may not be the technique of choice. In addition, there is no doubt that because most people have been taught frequentist statistical methods at school, more effort is required to switch over and adopt the simpler Bayesian mind-set.
Nonetheless, Bayesian inference is the generally applicable technique for extracting the most possible data from difficult, messy data sets. This article has shown why further acceptance of Bayesian methodologies could lead to significant improvements in diagnostic devices.
Roger F. Sewell, DM, is a senior consultant at
Cambridge Consultants Ltd. (Cambridge, UK). He can be reached at rfs@
Amanda J. Fuller, PhD,
is group leader of the diagnostic products group at Cambridge Consultants Ltd. (Cambridge, UK). She can be reached at amanda.fuller@
1. S McKee, “FDA Approves First IVDMIA,” IVD Technology 13, no. 3 (2007): 15.


Class 1 have started looking at a new text in English, ‘The Elves and the Shoemaker’. So far the children have considered some of the vocabulary in the text and the definition of some unfamiliar words.
In mathematics we are learning about addition and subtraction and how we can find related facts from given number facts.
In science the class are exploring everyday materials and sorting objects by what they are made of. As a challenge, Year 2 have considered some everyday objects and researched where the material they are made from is found.
In geography the class are looking at the local area and have used Google maps to label some aerial photographs of the area around the school.
In computing we are looking at how algorithms are used to program and how they are linked to sets of instructions.
In design technology the children have investigated how to best join fabrics by testing different methods from glue, staples, pins, and tapes. The children have created poppy designs to display and show what they have learnt about Remembrance events.


The Earth is at absolute rest
At college and university students are taught that the laws of Mechanics are valid in all inertial frames of reference in the same form, and that no frame is preferred and hence no absolute speed can be determined, no object can be said to be at rest absolutely.
On the other hand, simple considerations suffice to debunk such nonsense.
A car with a mass of m = 1000kg, at rest with respect to a road, is accelerated to a speed of v = 100km/h = 100 1000m/3600s = 27.78m/s. The kinetic energy 1/2 m v^2 of the car is
E0 = 1/2 1000kg ( 0km/h)^2 = 0
E1 = 1/2 1000kg (100km/h)^2 = 385.80kJ
E = E1 - E0 = 385.80kJ
At rest, speed and kinetic energy E0 are 0. At 100km/h the kinetic energy E1 is 385.80kJ. To increase the speed of the car we need at least E = E1 - E0 = 385.80kJ.
Astronomers say that the Earth is moving at about 108,000 km/h = 30m/s around the sun.
Our road is at rest in an inertial frame of reference S. Let the road be moved with respect to another inertial frame of reference S', such that in S' the car is being accelerated from 108,000km/h to 108,100km/h. Then, in S' we have:
E0' = 1/2 1000kg (108,000km/h)^2 = 450.000GJ
E1' = 1/2 1000kg (108,100km/h)^2 = 450.834GJ
E' = E1' - E0' = 834MJ
The energy value of gasoline is around 30MJ/l. Calculations in S' say that we need at least 27,8l or more than 3 gallons of gasoline to increase the speed of the car.
Everybody knows that that's not true. Real cars in the real world surely use a lot more energy than the minimum calculated with respect to S, but 3 gallons obviously are way too much.
Consequently, energy calculations are not correct in all inertial frames. Engineers and experimental physicists know that calculations work out correctly in Earth-fixed frames, where the earth is at absolute rest. In these frames speed values are absolute speed values and resulting amounts of gasoline correspond to reality.
Contrary to what physicists teach, some frames are preferred, and the preferred frames are at rest with respect to each other and with respect to the Earth. Absolute speed can be determined corresponding to absolute kinetic energy values and equivalent amounts of chemical energy. There is absolute rest and absolute movement.


- About Lerner
- Faculty & Staff
- News & Events
Professor of Business Administration
Office: 235 Lerner Hall
- Ph.D. - University of Pennsylvania, 1990
- M.S. - University of Pennsylvania, 1987
- B.S. - Shanghai Jiao Tong University, 1985
- Optimization, game theory, and applications.
- Supply chain and service operations management.
- Productivity analysis and performance evaluation.
- Board of Directors, APICS, The Association for Operations Management, 2012-2013
- Senior Editor, Production and Operations Management
- Advisory Board, American Journal of Operations Research
- Y. Xia, B. Chen, and P. Kouvelis, "Market-based supply chain coordination by matching suppliers' cost structures with buyers' order profiles", Management Science 54, (2008), 1861-1875.
- J. Yue, B. Chen, and M. Wang, "Expected value of distribution information for the newsvendor problem", Operations Research 54, (2006), 1128-1136.
- B. Chen, X. Chen, and C. Kanzow, “A penalized Fischer-Burmeister NCP-function", Mathematical Programming 88, (2000), 211-216.
- B. Chen and N. Xiu, “A global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing function", SIAM Journal of Optimization 9, (1999), 605-623.
- B. Chen and C.S. Lin, “Robust min-max regret one-median location problem on tree", Networks 31, (1998), 93-103.
- B. Chen and P.T. Harker, “Smooth approximations to nonlinear complementarity problems", SIAM Journal of Optimization 7, (1997), 403-420.
- B. Chen and P.T. Harker, “A non-interior continuation method for linear complementarity problems", SIAM Journal of Matrix Analysis and Applications 14, (1993), 1168-1190.
- D. Kraay, P.T. Harker and B. Chen, “Optimal pacing of trains in freight railroads: model formulation and solution", Operations Research 39, (1991), 82-99.
- K.W. Ross and B. Chen, “Optimal scheduling of interactive and noninteractive traffic in telecommunication systems", IEEE Transaction on Automatic Control 33, (1988), 261-267.


Present tutorial: This tutorial was prepared for our freshman engineering students using the student version of MATLAB, because symbolic computations are covered in almost no introductory textbook. We are pleased to make it available to the international community, and would appreciate suggestions for its improvement. MATLAB’s symbolic engine permits the user to do things easily that are difficult or impossible to do numerically.
The following topics are covered:
Conversion to numeric (“double”) variables for subsequent numeric calculations
Plotting of equations and functions
Solution of equations: graphical approximations, single equations, multiple simultaneous equations, non-linear equations, equations with infinitely many solutions.
Variable precision arithmetic
Powerpoint lectures and homework assignments
First do the demos in MATLAB’s help. Click on Help, Demos tab at the top, Toolboxes, Symbolic Math.
In the following instructions, material in bold is to be entered in the MATLAB command line and executed. This can be done by cutting and pasting using Ctrl c to cut and Ctrl v to paste. (Use the small finger on the left hand to press Ctrl while pressing c or v with the index finger of the left hand at the same time. Ctrl x cuts the highlighted material.)
The first step is to tell MATLAB that the variables to be used are symbolic. This is most easily done using the syms command (see >> help syms). For example, in the following we will use x, y, a, b and c as symbolic quantities. So enter >> clear, syms x y a b c f g . In the Workspace window (or use >> whos) note that these variables are symbolic, rather than the usual double precision numeric variables. It’s a good idea to start with “clear” to make certain MATLAB doesn’t use previous values for variables.
Differentiation: The MATLAB symbolic toolbox is very useful for checking calculus problems. For the syntax for symbolic differentiation, see >> help sym/diff . For example, to find diff_f = where f = e-axx3bsin(cx) and a, b and c are unspecified constants, we do the following1:
>> clear, syms a b c x; f=exp(-a*x)*x^(3*b)*sin(c*x), diff_f = diff(f,x)
Notice that the result is quite complex. To ask MATLAB to try to simplify it, type:
>> diff_f_simp = simple(diff_f)
(see >> help sym/simple). MATLAB uses several algorithms for algebraic simplification, and chooses as a final answer the shortest of the results. If you prefer some other form listed, select the command producing it.
MATLAB can take higher order differentials. For example, to find diff2_f =we do
>> diff2_f=diff(f,x,2) using the already defined expression for f.
Indefinite integrals: For the syntax for symbolic integration, see >> help sym/int. Thus, for example, to find the indefinite integral int_g =, where g = e-axsin(cx), we do
>> clear, syms a c x; g=exp(-a*x)*sin(c*x), int_g = int(g,x)
Check this by taking >>diff_int = diff(int_g,x). This should give back g, but it doesn’t look the same because MATLAB doesn’t always give things in the simplest format. In this case, we again use the simple command >> diff_int_simp = simple(diff_int).
Note that MATLAB does not add the constant of integration (“C”) customarily shown for indefinite integrals -- this you must do by adding C after the int command.
Sometimes MATLAB is not able to integrate an expression symbolically. For example, try to find (where, as before, f = e-axx3bsin(cx) and a, b and c are unspecified constants).
>> clear, syms a b c x; f = exp(-a*x)*x^(3*b)*sin(c*x), int(f,x)
Notice that when MATLAB cannot find a solution, it gives a warning message and repeats the command. When this happens, you’re out of luck.
Definite integrals: For the definite integral, int_def_g =do
>> clear, syms a c x; g = exp(-a*x)*sin(c*x), int_def_g = int(g,x,-pi,pi)
When MATLAB is unable to find an analytical (symbolic) integral, such as with >>int(exp(sin(x)),x,0,1), a numerical solution can nevertheless be found using the “quad” command, e.g. >> quad('exp(sin)',0,1) . Generally speaking, it is best to follow the steps below:
1.Create the function to be integrated in the MATLAB editor and save it to your Current Directory, which must also be in the path (File / Set Path). For example, for our exp(sin(x)) function:
function out = func1(x)
out = exp(sin(x));
2.Use either one of the following formats: >> quad('func1',0,1) or >> quad(@func1,0,1)
Conversion to a double precision numeric variable: Often MATLAB's symbolic engine produces a symbolic answer, as in:
>> clear, syms x; h=exp(-x)*sin(x), int_def_h=int(h,x,-pi,pi)
where is retained in symbolic form (pi). Irrational numbers may also result, e.g. 4/3 rather than 1.3333. For use in subsequent numerical calculations, you must convert symbolic results into numeric “double” variables by the “double” command:
Substitution: Sometimes it is desired to substitute one value or parameter for another in an expression. This can be done using the “subs” command (see >> help sym/subs). For example, imagine we want a=2 and c=4 in int_def_g. This is done by:
>> clear, syms a c x; g = exp(-a*x)*sin(c*x), int_def_g = int(g,x,-pi,pi)
Do not try to assign the values to the variables before the integration or differentiation is performed. (Try it in the above example and see what happens.)
Graphing equations and functions: MATLAB’s symbolic engine has commands for plotting symbolic equations and functions, appropriately starting with “ez.” For two-dimensional graphs, see >> help ezplot. As an example here we use the equation for a circle of radius 2, x2+y2 = 4.
Step 1: Move everything to the left-hand side of the equation, i.e. x2 + y2 – 4 = 0
Step 2: Clear and define x and y as symbolic.
Step 3: Define a variable equal to the left-hand side of the equation, inside single
straight2 quotation marks.
Step 4: Use ezplot.
>> clear; syms x y; lhs = 'x^2+y^2 - 4'; ezplot(lhs,[-3,3,-2.5,2.5])
The [-3,3,-2,2] tells MATLAB that the x scale is to go from -3 to +3, and the y scale is to go from -2.5 to +2.5. Note that we have assigned x2 + y2 - 4 to the character variable “eq” (see the Workspace window or type >> whos).
You can edit the figure in the Figure window. First save it as “circle.fig” using File/Save As so you can retrieve it later from the Command window, without having to recreate it. Here are some things you can do in the Figure window:
Click on the arrow icon and then double-click on any portion of the figure you’d like to edit. In this way you can change the color and thickness of the curve, add points, change the axes, edit the title.
Edit / Copy Figure to place it on the Windows Clipboard. Then you can paste it into a Word document, for example.
While in the Figure window, press the Alt and Print Screen keys at the same time, to place the entire figure window on the clipboard. This can also be pasted into a Word document.
Experiment with other equations.
Three-dimensional plots can be created using the ezmesh and ezsurf commands. For these commands an explicit form must be used, i.e. z = f(x,y). It is the f(x,y) that is entered into the command. The code below illustrates some possibilities. The green lines with % are comments and are not executed by MATLAB.
% Some 3d plots, where z = the expressions shown
% In the figure window, click on Tools, Rotate 3D.
% Then put cursor on figure, hold down left mouse,
% move cursor to rotate figure. Notice that this
% gives you a better feeling for its 3D shape.
% Have some fun and create your own shapes.
syms x y
ex4='x + y';
Open the excellent MATLAB editor by >> edit, cut and paste the above code into your editor, save it on your Current Directory as symplot3D.m, make certain the Current Directory is in MATLAB’s path (File/Set Path), and then execute it in the Command window by >>symplot3D.
Select each figure in turn, and in the Figure window Tools / Rotate 3D. Put the cursor anywhere on the figure, hold down the left mouse button, and move the mouse to rotate the figure.
Solution of equations: The symbolic toolbox in MATLAB is capable of solving many types of equations, including non-linear ones and several simultaneous equations. See: >> help sym/solve. The steps in solving one or more equations using “solve” are:
Step 1: Define the variables in the equations as symbolic using the “syms” command.
Step 2: Define the equations (see the examples below).
Step 3: Solve the equations using the “solve” command.
Step 4: If there is more than one solution, decide which is physically reasonable and select it.
Step 5: Check the solution by substituting it back into the original equation(s).
We now look at some examples.
Example 1: Find the two solutions to the classic quadratic equation, ax2 + bx + c = 0.
>> clear, syms x y z a b c; eq = 'a*x^2+b*x+c = 0', [x]=solve(eq,x)
(Note that single quotation marks must be placed around an equation that is assigned to a variable, here “eq”.) As expected, we obtain two solutions. Suppose we want only the first. This is extracted by >> x1 = x(1) . This should be done manually or using a logical operator (“if” command), because MATLAB's symbolic engine doesn't always give multiple solutions in the same order. What is x(1) now could be x(2) in a subsequent trial.
Example 2: Find the solution to e2x=3y:
>> clear, syms x y; eq = 'exp(2*x) = 3*y', [x] = solve(eq,x)
(Is the log in this solution the natural log or the log base 10? See
We use a similar procedure to solve simultaneous equations. First, let us look at a set of linear equations, and then non-linear equations.
Example 3: Find the solution to the following set of linear equations:
2x-3y+4z = 5
y+4z+x = 10
-2z+3x+4y = 0
>> clear, syms x y z;
>> eq1 = '2*x-3*y+4*z = 5'
>> eq2 = 'y+4*z+x = 10',
>> eq3 = '-2*z+3*x+4*y = 0'
>> [x,y,z] = solve(eq1,eq2,eq3,x,y,z)
Notice that we did not have to enter x, y and z in any particular order in defining the equation variables (unlike the numerical MATLAB method using matrices with \ or inv). However, the variables in the output are in alphabetical order, regardless of how they are written in the command. So take care to list the variables in the brackets in alphabetical order. (For example, if you use [z,x,y] rather than [x,y,z] in the above example the result produced for z is actually x, that for x is actually y, and that for y is actually z.) Again, the results may be symbolic and require conversion to numeric form for subsequent calculations:
>> x_d = double(x), y_d = double(y), z_d = double(z)
Example 4: We can also solve simultaneous non-linear equations, e.g. y=2ex and y=3-x2:
>> clear, syms x y; eq1='y=2*exp(x) ', eq2='y=3-x^2', [x,y]=solve(eq1,eq2,x,y)
Note that this gives only one solution; there are two as we see below. Again, to convert x and y from symbolic variables to numeric double variables, use the “double” command. (Use “whos” or look in the Workspace window to follow this transformation.)
We can obtain an approximate solution by plotting these two equations and looking at their intersections. As above, we first move everything to the left hand side of the equations.
>> clear, syms x y; eq1 = 'y - 2*exp(x)'; eq2 = 'y-3+x^2';
>> ezplot(eq1), hold on; ezplot(eq2); hold off
The "hold on" command tells MATLAB to write following plots on top of the first one, rather than replacing it. Approximate values of the solutions (intersections) can be obtained in the Figure window by clicking on Tools / Zoom In, clicking on a desired intersection to enlarge that area of the graph, clicking on Tools / Data Cursor, and then clicking on that intersection again to give the approximate values of x and y there. Compare these results with those found above using the solve command.
Example 5: There are equations with an infinite number of solutions, for example sin(x) = 1/2.
It is helpful to see some of these solutions by plotting y = sin(x)-1/2 and seeing approximately where it has values of zero:
>> clear, syms x; eq='y - sin(x) + 1/2'; ezplot(eq,[-6,6,-2,1])
>> hold on; eq0='0'; ezplot(eq0); hold off
The "hold on" command tells MATLAB to write following plots on top of the first one, rather than replacing it. Plotting 0 puts a horizontal line on the graph. Intersections of the sine wave with this line represent solutions of the first equation. Approximate values can be obtained in the Figure window by clicking on Tools / Zoom In, clicking on a desired intersection to enlarge that area of the graph, clicking on Tools / Data Cursor, and then clicking on that intersection again to give the approximate values of x and y there. Try this on the first two intersections to the right of the origin (x > 0). Write down your results to compare with those found below using solve commands.
The symbolic solve command finds only one of these solutions and then stops:
>> clear, syms x; eq = 'sin(x)-1/2=0'; [x] = solve(eq,x); x_doub=double(x)
But suppose we want the solution that lies between x = 2 and x = 3. There are two ways to do this, one using MATLAB's symbolic engine and the (easier) using its numeric engine. First we use the symbolic fsolve command (see >> mhelp fsolve), for which the syntax is
>> clear, x = maple('fsolve(sin(x)=1/2,x,2..3)')
While not immediately apparent, >> whos shows that the result is a 1x33 character array (made up of the ASCII decimal codes for each number and the period.) Try >> x_d = double(x). To convert to a number that can be used in subsequent calculations, we must use the “str2num” (convert string to number) command:
>> x_d = str2num(x)
(Unfortunately, there are many symbolic commands that are not included in the Student version of MATLAB. These can be recognized by the use of “mhelp” and “maple.” For these, you’ll have to use the full professional version of MATLAB that is installed on Clarkson’s computers.)
Now for an easier numerical method to solve the above problem. See >> help fzero. Here are the steps involved:
1.Convert the equation to a function consisting of everything moved to the left-hand side, e.g. func2(x) = sin(x)-1/2.
2.Create this function in the MATLAB editor, save to the Current Directory, and make certain the current directory is in the path (File / Set Path). For example:
function out = func2(x)
out = sin(x) - 1/2;
3.Plot this function over the range of interest to see where the solutions are, e.g.:
>> clear, x = - 4:0.01:4; plot(x,func2(x))
4.Use one of the following formats to find the solution:
>> fzero('func2',3) or >> fzero(@func2,3) or >> fzero('func2', [2,3])
MATLAB finds the value of x near 3 that gives func2 = 0. Notice (“whos” or Workspace) that the result is double, so no conversions are necessary for subsequent numeric calculations.
Limits: The symbolic toolbox also contains a command for finding limits of expressions, e.g.:
>> clear, syms x a; value = limit(sin(a*x)/x,x,0)
There are cases where the limit depends on the direction from which it is approached. Consider the plot of tan(x) versus x generated by:
Suppose we want the value of tan(x) at x = p/2. We could use MATLAB's numeric engine as follows:
Does this give a correct result? Recall that tan(p/2) can be either + or -, depending whether we approach from the left or the right. The numeric solution is a very large number, which might by okay for some applications but not for others. We can use the symbolic engine to find the two correct solutions, as follows:
>> tanhalfpiright = limit(tan(x),x,pi/2,'right')
>> tanhalfpileft = limit(tan(x),x,pi/2,'left')
As an exercise, find . The result may surprise you!
Taylor series: Taylor series are polynomial approximations for functions that are valid near a particular point. Differentials of increasingly higher order are used to generate this polynomial (e.g., see your calculus test). The syntax is taylor(expression, order, variable, point)
Example: Expand ex about x = 3 to order 5:
>> clear, syms x; Tay_expx = taylor(exp(x),5,x,3)
This is exactly correct only at x = 3. Compare this approximation at x = 2 with the exact value:
>> approx=subs(Tay_expx,x,2), exact=exp(2), frac_err=abs(1-approx/exact)
The approximation improves as one adds more terms to the polynomial (order), or moves closer to the point about which the expansion was made. This can be illustrated by:
>> ezplot(Tay_expx), hold on, ezplot(exp(x)), hold off
Summations: MATLAB’s symbolic engine is also capable of doing symbolic summations, even those with infinitely many terms! The “symsum” command is used (see help symsum) Let us consider, where f(k) is a function of an integer k. The syntax to calculate this using
MATLAB is syms k, symsum(f,k,m,n). If a factorial (!) is desired, since ! is also a numeric command, we must tell MATLAB to use the symbolic form by including the term involving in the factorial, say (1+2*k)!, in sym, i.e. sym('(1+2*k)!'). Infinite sums are particularly interesting, e.g.: , the solution for which is found by:
>> clear, syms k x; symsum((-1)^k*x^(2*k+1)/sym('(2*k+1)!'),k,0,Inf)
As an exercise, find and. The results may surprise you!
Variable Precision Arithmetic (VPA): We can control the precision with which MATLAB performs calculations involving irrational numbers. We illustrate this using the golden ratio:
Read >> help vpa and >> help sym/vpa to learn about the two ways of setting the precision of calculating an irrational number. What is the current (default) number of digits?
>> clear, digits
This is the default accuracy of MATLAB's numeric engine. The VPA syntax for calculating R from S is: R = vpa(S) & R = vpa('S'), depending on whether you want the answer accurate to about 16 significant figures or to full precision in symbolic form. Find the difference between the results of calculating the golden ratio by these two methods.
>> phi1 = vpa((1+sqrt(5))/2), phi2 = vpa('(1+sqrt(5))/2'), difference = phi1 - phi2
Which is closer to the correct irrational value? When "difference" is evaluated, MATLAB takes phi2 and expresses it to the current number of digits. The number of digits to which the answer is rounded off can be controlled by the optional argument D, i.e. vpa(S,D) or vpa('S',D). Compare the following:
>> clear, phi1 = vpa((1+sqrt(5))/2, 4), phi2 = vpa('(1+sqrt(5))/2'), difference = phi1 - phi2
>> clear, phi1 = vpa((1+sqrt(5))/2), phi2 = vpa('(1+sqrt(5))/2',4), difference = phi1 - phi2
>> clear, phi1 = vpa((1+sqrt(5))/2,100), phi2 = vpa('(1+sqrt(5))/2',100), difference = phi1 - phi2
Thus, increasing D beyond 32 does not improve the precision using the numerical format, but does add more significant figures using the symbolic format (with the quotation marks). So if you want to know an irrational number to more significant figures, use symbolic methods.


Suntex International 24 Game: Fractions/Decimals
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Double-sided, 4'' x 4'' cards. Deck contains 48 Fractions 1 Dot, 2 Dot and 3 Dot cards, 48 Decimals 1 Dot, 2 Dot and 3 Dot cards. Three levels of challenge. Cards and offer three levels of difficulty. Easy, 1 Dot cards are easy, 2 Dot cards are medium and 3 Dot cards are tough. How to play Fractions/Decimals. Object of the game is to make 24. You can add, subtract, multiply and divide. Use all four numbers on a card, but use each number only once. There is at least one solution to every card.
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Table of contents
- Environmental Monitoring and Characterization
- 250 Problems in Elementary Number Theory - Sierpinski (1970).pdf
- Timeline of number theory - Wikipedia
- Elementary Number Theory
Environmental Monitoring and Characterization
Your input will be highly appreciated. The lecture notes and handouts from these training camps provide a rich source of preparatory material and problems for the Mathematical Olympiad. Several noted mathematicians and previous champions of the Olympiads contribute to these materials regularly.
The students are encouraged to consult the following resources. Art of Problem Solving AoPS offers an amazing online community of likeminded students and educators who enjoy discussing interesting mathematical problems.
You may find AoPS community and their resources extremely useful while preparing for the Olympiads. Please note that AoPS may hold quick answers to many mathematical problems -- but the goal for you should be to arrive at the solution by yourself -- even if it is really painstaking. So, have fun discussing problems, but please avoid any shortcut to problem-solving.
An Excursion in Mathematics Editors: M. Modak, S.
250 Problems in Elementary Number Theory - Sierpinski (1970).pdf
Katre and V. Krishnamurthy, C. Pranesachar, K. Ranganathan and B.
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Elementary Number Theory
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A+ Test Prep & Tutoring helps students get to the next level. Sometimes that means preparing a student to earn the few extra SAT points he needs to be accepted into his dream school. However, we also help students with a broad range of learning disabilities to prepare for the SAT and ACT. As a parent, you may feel stress, anxiety, and guilt about your student’s learning issues—but you’re not alone. Whether you live near our Gwynedd or Jenkintown location or prefer in-home sessions, our tutors can bring your student the specialized attention he needs to succeed. The following are some reasons that A+ Test Prep & Tutoring’s one-to-one services may be particularly beneficial to students with special learning needs.
A+ Test Prep & Tutoring Blog
Posts about Philadelphia Tutor
This week we’d like to spotlight A+ tutor Tom Spencer. Tom is a software development management consultant, with a bachelor's degree in math from Bucknell University.
Tom works with our ACT and SAT prep students on the math portions of the test. He also tutors our Philadelphia-area students in Algebra, Calculus, Geometry, Probability & Statistics, and Trigonometry.
McLean Dyer is the kind of person one thinks of when the phrase “practice makes perfect" is used.
Though McLean is from Wayne, Pennsylvania, he’s attending high school at Saddlebrook Tennis Academy, a premier tennis academy in Wesley Chapel, Florida.
This week we'd like to spotlight A+ tutor Bob Wolpert. Bob, a full-time math teacher for more than 40 years, who specializes in geometry and algebra tutoring for our Philadelphia area tutoring students. He also works one-to-one with our ACT and SAT test prep students on the math sections of the college admissions exams.
All of our Philadelphia tutors love working with our students on a day-to-day basis. To get an idea of why they love doing what they do, we decided to ask them at a recent professional development conference.
Last week A+ Test Prep and Tutoring president Daniel Ascher, M.Ed. was featured in Philadelphia Family Magazine's Asked and Answered article, "When Is It Time to Seek a Tutor?"


Since 1993 the Ingenuity Project has worked in collaboration with the Baltimore City Public School System to provide Baltimore's brightest middle school students with a free, highly accelerated, and challenging mathematics and science curriculum.
The Ingenuity Project®, a non-profit organization, is a joint effort of the Baltimore City Public School System, the Abell Foundation and Baltimore's science and mathematics community. Our mission is to prepare highly capable and motivated students in Baltimore City to achieve at nationally competitive levels in mathematics, science, technology, and related fields. Graduates of the Ingenuity Project are highly sought after by the most elite schools and colleges.
Our middle school program is a full-time citywide program offered to academically gifted city students at Roland Park Middle School #233, Mount Royal Middle School #66, and Hamilton Middle School #236. Ingenuity offers advanced mathematics and science classes during the school day.
Our high school program is a full-time citywide program offered to academically gifted students at the Baltimore Polytechnic Institute. Ingenuity high school students take advanced placement courses, work with scientist mentors to develop independent research projects, and submit their work to national competitions.
Read more on the blog:
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A quantum-mechanical problem of a pair of identical particles interacting with a third particle via zero-range interactions is discussed.
A complete description is given to a system of two identical fermions (bosons) and a third particle. Correct formulation of the problem is presented, and bound-state energies are calculated for various total angular momenta. For this purpose, the generalized Coulomb problem is analyzed and its correspondence to the three-body problem is used to introduce the boundary condition in the vicinity of the triple-collision point. The related mathematical aspects and remaining obscure points are discussed.
Exact diffractionless solutions are obtained for the system of two-component one-dimensional particles with zero-range interactions. A new three-body bound state of odd parity is found, and the region of the parameters for which it exists is identified. One of the boundaries of this region is exactly determined by a diffractionless solution.
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Mini-Enrichment Courses: STEP offers courses in science
and mathematics which expand upon and enhance students'
educational experiences and develop their critical
thinking and reasoning skills. Examples of mini-courses
are SAT Review and Regents Review classes.
Saturday Workshops: On Saturday
mornings, STEP provides hands-on activities, presentations,
and experiments which actively engage students in
the study of the sciences.
Tutorials: STEP provides students
with regular assistance in school work on a group
and on an individual basis.
SAT Review: STEP provides students,
Fall and Spring semester, with a thorough program
in verbal and mathematics. A comprehensive review
is also given twice a year during a Saturday Workshop
Regents Review: A complete regents
review in mathematics and science is provided.
Counseling & Needs Assessment:
STEP offers academic and career counseling services,
as well as evaluation of student strengths and needs.
Engineering Technology Projects
Field Trips: To reinforce learning
and interest in scientific areas and to explore academic
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are taken to such sites as museums, science centers,
and major universities.
Cultural Activities: STEP provides
trips to plays, dance productions, and other cultural
events, to enhance the students' appreciation of these
aspects of culture, especially in the areas relative
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Academic Camp: STEP offers a more intensive
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Students from across the city came to the Intrepid on Thursday to take part in a bundle of science, technology, engineering and math or STEM activities as part of a celebration for Time Warner Cable's Connect a Million Minds initiative.
The celebration brought Time Warner Cable executives and local non-profits together to commemorate the initiative reaching a million mind milestone, connecting one million young people to science, technology, engineering and math.
Students got to share their love for STEM and presented the work they have been doing all summer with their organizations.
Executives say while they have reached one million minds, it's not stopping there.
"No question that this is not the end, it’s another milestone along the way, and we've changed our mantra to connecting a million minds and counting. So it keeps going from here," said Time Warner Cable Chairman and CEO Rob Marcus.
Time Warner Cable is NY1's parent company.
To find hands on science, technology, engineering and math opportunities in your community visit connectamillionminds.com.


Welcome to the International Conference on Mathematics and Its Applications
From July 14 to 15, 2022 the International Conference on Mathematics and Its Applications (ICMA 2022) will be held at Ton Duc Thang University (TDTU), Ho Chi Minh City, Vietnam, along with Department of Mathematics - Incheon National University (Korea), Department of Mathematics and Statistics - Mississippi State University (USA), Department of Mathematics - University of Louisiana at Lafayette (USA), Faculty of Mathematics and Computer Science - University of Science, Vietnam National University - Ho Chi Minh City (Vietnam).
ICMA 2022 provides a forum for university academics, scholars, researchers and graduate students around the world to share their knowledge and latest results in Pure and Applied Mathematics, Probability and Statistics. The conference will take place in a hybrid mode. It will include participations both online (mainly for the international participants) as well as in-person (primarily for the local participants).


6th and 7th Grade math classes are working on final semester study guides today Friday, December 12, and will continue on Monday, December 15. Students should be studying sections they struggled to complete.
The semester exam for Math classes will start on Tuesday and continue to Wednesday. If you have any questions please feel free to email me.
Have a wonderful weekend.
Previously I have listed lesson plans weekly. I am currently working on a link to a Google calendar so this is easier to work with. Thanks….


When students understand mathematics, they are equipped with knowledge they can bring to every aspect of their lives.
Choose a Path that includes Math is a comphrehensive, user-friendly online tool kit for Ontario Students, their Parents, and Teachers. This resource shows us WHY selecting high school math courses is so important, and HOW we can all work together to support 'math success' and bright futures for our youth!
Math Storytime is an engaging interactive storybook app for children ages 5 to 7, and beyond. Each wonderfully illustrated story features a unique take on a math theme, and allows users to choose to read themselves, or be read to. Parents and caregivers can also engage their children in fun activities and 'math talk¹ through discussion bubbles which extend the learning within each story.
When students understand mathematics, they are equipped with knowledge they can bring to every aspect of their lives. Inspiring Your Child to Learn and Love Math is a tool kit for parents. It provides modules with simple, but effective methods and materials for parents.


Devoxx UK 2019
from Wednesday 8 May to Friday 10 May 2019.
Roy van Rijn is director of OpenValue Rotterdam and software developer at JPoint, Java Champion, JavaOne Rockstar, public speaker, blogger, loves: agile, cryptography, math, algorithms. He tweets (@royvanrijn) and blogs: http://royvanrijn.com
See also https://royvanrijn.com
How do quantum computers work? What is a qubit? What the hell happened to Schrödinger's cats? During this talk we'll together unravel the mysteries of entangled qubits. We'll explain the basics of quantum computing in simple and fun ways, including visual demonstrations with polarsation filters and a skippyball (known as Mr Qubit).
We'll even build an actual working quantum computer.... tribute, a simulator, in Java!
And I'll explain how larger quantum algorithms work by explaining the quantum benefits in Shor's Prime Factoring algorithm.
But beware, this talk isn't for the faint hearted. We'll keep away from all the unnecessary complicated math, but if Math.random() is unlucky we might kill some more cats.


I have a statistics problem that I don’t know if I am solving correctly. I am asking the internet to either verify my work or correct it.
Here’s the problem:
What are the odds of dealing a specific sequence of 18 cards from a standard, randomized 52 card deck? Stated another way, there are 9 poker players sitting around a table. In the game of Texas Hold’em each player is dealt 2 cards. The dealer deals each player one card and then starts over again dealing the second. What is the probability a specific order of cards (both in value and in suit) is dealt? Of course the situation doesn’t matter since it is really just the first 18 cards out of the deck but I thought I would clarify the question. The reason why I want to know this will be discussed AFTER the answer is determined.
Here is my solution:
The odds of any one card coming out 1st is 1 out of 52. The odds for the 2nd card are 1 out of 51. This goes on until all 18 cards are dealt. So, the math is simply (in my thinking at least) is 52 * 51 * 50 * 49…all the way to 35. This gives an answer of 2.73 x 10^29 to 1 against.
Am I close?
And just FYI…there reason I want to know the answer has nothing to do with poker or gambling. I am just using the card example to help make a point about something entirely unrelated.


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Play this game and get math skills by overcome the levels.
Add numbers until you get the target result. Start with basic goals and improve your math skills by overcoming the difficulty levels.
The game is specially designed for you to acquire mental agility when adding numbers.
Play on two different modes: Timer and Infinite
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