text:
In the present analysis several simulation analysis were carried out to analyze the structural response of CF / PEEK laminated composites showing great potential for application in the biomedical field.
Background, in a laminated composite the structural behavior depends on the arrangement of the individual layers, in these analyzes it is investigated how the different layouts influence the response of some structural elements;

Results, different simulation campaigns show the variation of the stiffness and of the coupling conditions between the load conditions and the structural response to the variation of the layout and the grain angle of the foil;
Conclusions, structural responses are highly variable, from simpler behaviors to predict to more complex responses. An analysis that takes into account the layout of the entire laminate, starting from the composition of the single layer, allows for precise results to understand the structural behavior.

The following report is extracted from a project of the Computational Mechanics of Tissues and Biomaterials course, held for Medical Engineering at the University of Rome Tor Vergata.

Table of Contents

## Introduction

The purpose of the following analysis is to investigate the behavior of different layouts of a laminated composite. A CF / PEEK laminate composite is analyzed, which is a composite with a thermoplastic polymer matrix reinforced with carbon fibers that shows great potential for biomedical applications.

Two fundamental structural elements are considered, a cylinder and a plate, and it is studied how the variation of the layout leads to a different mechanical response. An in-depth study is also carried out by analyzing the distribution of stresses at the level of the individual layers.

The knowledge and the possibility of predicting the structural behavior starting from the arrangement of the layers is fundamental both for the correct simulation and for the production of devices and objects with laminated composites.

## CF/PEEK laminates

Recent advances in composite materials, in particular with a PEEK matrix, or a polymeric matrix of poly-ether-ether ketone, have greatly expanded the application scenarios by making these materials fall into the class of technopolymers. Given its properties, this polymer ranks among the most important polymers in the engineering field. It has excellent properties such as high mechanical resistance, excellent thermal stability, chemical resistance and stable behavior even in chemically hostile environments. It also has an anticorrosive nature and good resistance to degradation, properties that make it excellent for applications in the biomedical field.

In these applications, some surface properties have also been improved by combining PEEK with bioactive particles such as hydroxyapatite. This has made it possible to address some problems such as the limited effectiveness in making cells adhere and in integrating with bone. Clearly the integration of reinforcing fibers has also allowed the improvement of the mechanical properties.

One of the most important reinforcements is that of unidirectional carbon fibers. Indicated as CF / PEEK, it was introduced around 1980 and in recent years it has proved to be an excellent biomaterial with great potential for application in biomedical systems [1]. Generic PEEK find great applications both in customized plants made with 2D printing and in orthodontics. Furthermore, CF / PEEK finds great applications in orthopedics. It should also be noted that continuous fiber CF / PEEK exhibits better mechanical properties than generic or short fiber reinforced PEEK but requires more advanced manufacturing methods.

### Biomedical applications

Typically in orthopedics, metal implants are used with a stiffness of about an order of magnitude higher than that of physiological bone and this can lead to problems related to bone resorption. In addition, there may be problems related to CT or X-ray diagnostic imaging [2].

The use of composite materials can provide several advantages such as the presence of anisotropic properties, radio-transparency, high fatigue strength and a high strength / weight ratio. Currently, there are still very few laminated composites actually used in orthopedic applications. An example of this material is PEEK reinforced with continuous carbon fiber, known as PEEK-OPTIMUM Ultra-Reinforced, and has been approved for implantation in humans by the American Food and Drug Administration [3].

The following simulation campaign refers to a composite laminate with CF / PEEK layers with a fiber volumetric fraction of 62% and considers different laminated layouts by analyzing the mechanical performance and the structural response.

## Background

Typically in orthopedics, metal implants are used with a stiffness of about an order of magnitude higher than that of physiological bone and this can lead to problems related to bone resorption. In addition, there may be problems related to CT or X-ray diagnostic imaging [2].

Modern composites are based on the use of a reinforcement phase, typically in the form of fiber, immersed in a matrix. Usually a polymeric matrix is used and glass, carbon, aramid and others can be used as reinforcing fibers. There are several methods of arranging the fibers and their distribution influences the overall behavior.

By putting together several layers of composite material, a package is obtained which is called a laminated composite.

The problem can be divided on three different scales of magnitude: the behavior of the final structure (macroscale) which depends on the laminate which in turn is influenced by the single laminae (mesoscale). In turn, the behavior is determined by the individual constituents and their composition (microscale). A schematic representation is given in fig. 1.

The following simulation campaign refers to a composite laminate with CF/PEEK layers with a fiber volumetric fraction of 62% and considers different laminated layouts by analyzing the mechanical performance and the structural response.

### Mixtures rule

The single layer is composed of a matrix and reinforcing fibers with a precise volume fraction. The first step is to homogenize the material properties of the single layer. We consider a grain such as to ensure transversely isotropic symmetry and the axis of the fiber with isotropic direction.

To get to the homogenized properties of the single layer it is possible to apply the rule of blends [6].

Considering a global volume of the composite given by the sum of the volume of the matrix and that of the reinforcement and assuming perfect adhesion between the two, it is possible to analytically calculate the resulting modules. The volumetric fractions apply:

$$v_{m}+f_{f}=1$$

And it is possible to express the resulting material properties as a function of the volumetric fraction of fibers (ff) and the material properties of the individual constituents, as expressed in eqs. (2) to (6).

$$E_{1}=f_{f} E_{f}+\left(1-f_{f}\right) E_{m}$$
$$E_{2}=E_{3}=\left(\frac{1-f_{f}}{E_{m}}+\frac{f_{f}}{E_{f}}\right)^{-1}$$
$$G_{12}=\left(\frac{v_{m}}{G_{m}}+\frac{v_{f}}{G_{f}}\right)^{-1}$$
$$v_{12}=v_{f} v_{f}+v_{m} v_{m}$$
$$v_{23}=\frac{E_{2}}{2 G_{23}}-1=\frac{E_{2}}{2 G_{12}}-1$$

It is possible to consider a compliance matrix of the type in eq. (7)

$$\left[\begin{array}{cccccc} \frac{1}{E_{1}} & -\frac{v_{21}}{E_{2}} & -\frac{v_{31}}{E_{3}} & 0 & 0 & 0 \\ -\frac{v_{12}}{E_{1}} & \frac{1}{E_{2}} & -\frac{v_{32}}{E_{3}} & 0 & 0 & 0 \\ -\frac{v_{13}}{E_{1}} & -\frac{v_{23}}{E_{2}} & \frac{1}{E_{3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G_{23}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G_{13}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{12}} \end{array}\right]$$

In particular, it is possible to observe the trend of the Young-type modules in fig. 2. The linear trend of E1 is evident. All the modules increase as the ff increases as the contribution of carbon fibers that have higher modules than PEEK increases. The following analysis considers an ff set at 62%.

### Thin laminate theory

After having homogenized the single layer, the theory of thin laminates [7] is used, treating the single layer as a homogeneous thin sheet. Each lamina is subjected to a plane state of tension and the stiffness matrix in the single layer can be expressed with respect to a global reference system.

$$[\bar{Q}]=\left[T_{\sigma}\right]^{-1}[Q]\left[T_{\epsilon}\right]$$

Considering the deformations in the plane of the mean surface and the curvature of the mean surface we arrive at the generalized constitutive laws, eq. (9).

$$\left[\begin{array}{c} N_{x} \\ N_{y} \\ N_{x y} \\ M_{x} \\ M_{y} \\ M_{x y} \end{array}\right]=\left[\begin{array}{llllll} A_{11} & A_{12} & A_{16} & B_{11} & B_{12} & B_{16} \\ A_{21} & A_{22} & A_{26} & B_{21} & B_{22} & B_{26} \\ A_{61} & A_{62} & A_{66} & B_{61} & B_{62} & B_{66} \\ B_{11} & B_{12} & B_{16} & D_{11} & D_{12} & D_{16} \\ B_{21} & B_{22} & B_{26} & D_{21} & D_{22} & D_{26} \\ B_{61} & B_{62} & B_{66} & D_{61} & D_{62} & D_{66} \end{array}\right]\left[\begin{array}{c} \varepsilon_{x}^{0} \\ \varepsilon_{y}^{0} \\ \gamma_{x y}^{0} \\ \kappa_{x}^{0} \\ \kappa_{y}^{0} \\ \kappa_{x y}^{0} \end{array}\right]$$

Where the three blocks of the matrix have a precise physical meaning. Block A, eq. (10), describes the mebranal stiffness of the single lamina. Block B, eq. (11), links the coupling between the membrane and flexural effects. Block D, eq. (12), represents the flexural stiffness of the laminate.

$$A_{i j}=\sum_{k=1}^{N} \bar{Q}_{i j}^{k}\left(z_{k}-z_{k-1}\right)$$
$$B_{i j}=\frac{1}{2} \sum_{k=1}^{N} \bar{Q}_{i j}^{k}\left(z_{k}^{2}-z_{k-1}^{2}\right)$$
$$D_{i j}=\frac{1}{3} \sum_{k=1}^{N} \bar{Q}_{i j}^{k}\left(z_{k}^{3}-z_{k-1}^{3}\right)$$

This description will allow to predict the couplings and the structural behavior starting from the structure of the three blocks.

### Properties of constituents

Starting from the single layer, a PEEK polymer matrix is considered which has a relatively low elastic modulus of 4 GPa with Poisson’s ratio ν = 0.36. The matrix is reinforced with carbon fibers which instead have a high stiffness, elastic modulus of 236 GPa and a Poisson’s ratio of 0.2. More information in tab 1.

The results of the mixture rule considering a volumetric fraction of fibers of 62% are shown in table 2. Starting from these numerical results, the stiffness matrix of the single layer is calculated.

We then proceed to consider different laminate configurations. Every single layer is considered with the fibers aligned along a main direction, with the exception of the ± 45f layers which are considered as a weave of fibers. The individual layers are then rotated with respect to a global reference and put together to form the composite laminate. Then it is possible to calculate the generalized constitutive bond of the laminate.

## Laminate layout

The variations of the laminate layout are infinite, it is possible to vary the number of layers and the grain angle. In the following analysis, 4 different cases are addressed:

\begin{aligned}
(a)& \:\left[\alpha /-\alpha / 30 /-30 / 0_{2}\right]_{\mathrm{s}} \operatorname{con} \alpha \in\left[0^{\circ} ; 90^{\circ}\right] \\
(b)&\: \left[-45 / 45 /-45 / 45\right]\\
(c)&\:\left[\pm 45^{f} / \pm 45^{f}\right]\\
(d)& \:\left[-30 /-45 /-30 /-45\right]
\end{aligned}

A representative illustration of the different layouts is shown in fig. 3. Two different structural elements are considered:

• Plate like element
• Cylinder like element

For the analysis of the structural response, different types of load are considered. On the plate type element, either an axial load, or a transverse load or both is considered. For the cylinder-type element, a uniform internal pressure is instead considered.

### Plate element

For the plate-like element, the four different laminate layouts are analyzed.

#### Case(a)

The first analysis deals with a parametric study (parameter α) on a layout of the type:

$$\left[\alpha /-\alpha / 30 /-30 / 0_{2}\right]_{S}$$

Three different load situations are considered: axial load only, transverse load only and both loads. The results are reported in figs. 5 to 7. As the parameter α varies between 0 ° and 90 ° different structural responses are observed.

Being a symmetrical laminate, a coupling between extensional and bending effects is not expected. This is confirmed by the comparison between the three cases. It is evident that by applying only an axial load, the flexural effect is much reduced compared to the other load configurations. In the case α = 0◦ the flexion is the minimum possible. Asα grows, the flexion effect increases. This does not happen with only the axial load where the bending effect is very low and practically zero both for α = 90◦ and for α = 0◦. In the latter cases, the coupling between in-plane and out-plane effects is therefore also null.

##### Simulation results

With the exception of the cases α = 90◦ and α = 0◦ there is always a torsional effect even if very limited.

It is evident that the arrangement of a greater number of layers aligned along the axis provides less flexural stiffness and this is reflected in a more pronounced flexural effect. In order to reduce the flexural effect it is good to arrange the parametric layers with the fibers orthogonal to the axis of the laminate. This also cancels the torsional effect and the in-plane out-plane coupling. In addition, it also provides greater membrane stiffness. Instead, if you want to have a more pronounced flexural effect, you can arrange the largest number of layers with the fibers aligned along the axis.

This leads to a shift in the free terminal side of the plate up to 5 times higher but makes the coupling with the torsional effects more evident. It is interesting to see that with only the transverse load, depending on the configuration of the laminate, it is possible to have such a bending as to move the final end both upwards and downwards (fig. 7 (iii)). This effect is not present in the cases α = 0 ° or α = 90 °.

Intermediate cases provide intermediate structural behaviors between the two borderline cases described. The coupling between the axial load and the torsional effect is maximum for the layout with α = 20 ° and α = 10 °.

#### Case (b)

A balanced laminate with layout is analyzed:

$$[-45 / 45 /-45 / 45]$$

Again the three different load configurations are considered.

Being a balanced laminate, a shear-extension fit is not expected. From the results obtained, figs. 11 to 13, several couplings are observed. A coupling is observed between the axial load and a bending effect. A coupling is also observed between the axial load and the torsional effect while a torsional effect is not observed with only the transverse load.

The behavior of this laminate is very similar to the behavior of the case (§4.1.1) with the parameter α = 90 ° but with a much lower bending stiffness due to the smaller number of layers.

#### Case (c)

A layout is analyzed:

$$\left[\pm 45^{f} / \pm 45^{f}\right]$$

The single layer is made up of an intertwined layer, that is a particular configuration where the fibers are crossed with a direction of ± 45◦. In the case of woven fabrics, the definition of the stiffness matrix of the layer changes:

$$Q_{i j}^{\text {woven }}=\frac{1}{2}\left[\left(\bar{Q}_{i j}\right)_{45}+\left(\bar{Q}_{i j}\right)_{-45}\right]$$

There is no torsional effect in this layout. There is also no coupling between axial loads and bending effect. The laminate responds without showing torsional or extensional-flexural couplings also due to symmetry and balance.

#### Case (d)

The following layout is analyzed:

$$[-30 /-45 /-30 /-45]$$

The numerical results are presented in figs. 11 to 13 compared with the case §4.1.2.

This layout is completely unbalanced and asymmetrical with the grain direction showing negative angles with respect to the laminate axis. This is reflected in the coupling of the torsional effect with all load conditions.

There is also the in-plane out-plane coupling as can be seen from the bending induced by the axial load alone (fig. 13 (iii)). Extensional-extensional coupling is also very marked.

### Data analysis

It is interesting to see the effect of the transverse load only (fig. 12) in which, due to the imbalance of the grain directions and torsion, there is an overall effect whereby the free end of the plate tends to have a displacement in the opposite direction to that of the load. This effect, in the case in which both load conditions are present, results in an initial raising and then flexing according to the direction of the transverse load (fig. 11 (iii)).

The complete deformation is also shown in fig. 14. This coupling results in a bending effect by applying only the longitudinal load (fig. 13 (iii)).

### Cylinder like element

A cylinder-type structural element with radius R and length L is then considered. A load configuration expressed by a uniform internal pressure on the cylinder wall is considered. The four different laminate layouts are analyzed again. The results are read on the edge of the cylinder in the four segments obtainable at x = ± R and y = ± R, so as to be able to understand if the widening is uniform or not along the axis and if there is a torsional effect.

#### Case (a)

A parametric simulation campaign is carried out and the structural response to the variation of the parameter α is analyzed. The results are reported in fig. 15. The layout of the laminate is of the type:

$$\left[\alpha /-\alpha / 30 /-30 / 0_{2}\right]_{S} \quad \operatorname{con} \alpha \in\left[0^{\circ} ; 90^{\circ}\right]$$

It is evident that as α increases, the cylinder becomes less and less compliant, that is, with the same load it expands less and less. The expansion is uniform and symmetrical, both along y ^ and along z ^. The greater number of fibers oriented parallel to the load direction (α → 90◦) makes the material more rigid and therefore reduces the extensional effect. Furthermore, being a symmetrical and balanced layout, no twisting effects are observed either.

#### Case (b)

A balanced laminate with the following layout is analyzed:

$$[-45 / 45 /-45 / 45]$$

In this layout, a uniform and symmetrical expansion of the cylinder is observed. The circular section widens and remains constant for the entire length of the cylinder. Furthermore, there are no torsional effects. The results are reported in fig. 16

#### Case (c)

The following layout is analyzed:

$$\left[\pm 45^{f} / \pm 45^{f}\right]$$

The results are shown in fig. 17.

The intertwined layout provides a symmetrical and balanced layout overall.

A uniform and constant dilation is observed as in the previous cases. Numerically, a slight difference is obtained by measuring the vertical displacement (along z) of the upper and lower edges, but it is negligible and less than 10–5. A lower stiffness than in the case §4.2.2 is also observed due to the smaller number of overall layers.

#### Case (d)

The following layout is analyzed:

$$[-30 /-45 /-30 /-45]$$

The results are shown in fig. 16 and a three-dimensional deformation is shown in fig. 4 (iv) The layout is highly unbalanced and although the dilation is uniform, a torsional effect is observed. Overall, the cylinder tends to screw.

This can be observed by reading the horizontal displacement (along y) of the lower and upper edge of the cylinder ( z = ± R ). As reported in fig. 18 an increasing linear displacement is observed along the axis of the cylinder. It is positive for the bottom edge and negative for the top edge.

## Plate thickness analysis

In the theory of thin laminates the strain field is reduced to a stain field of the average surface linked to sliding and expansion in the plane. Added to this are out-of-plane effects related to the deflection of the average surface and any torsion contributions. This implies that the deformation field is assumed to be constant and identical in all layers.

Let’s consider a further case with 12 layers having a layout like:

$$\left[10 /-10 / 30 /-30 / 0_{2}\right]_{\mathrm{s}}$$

A schematic representation is shown in fig. 19 (i).

The only thing that distinguishes one layer from the other, according to the considered theory, is the only coordinate z . So we can consider the reduced bond in the plane where each layer will have its constitutive aspect while the deformations will be related to the z coordinate.

$$\left[\begin{array}{c} \sigma_{x} \\ \sigma_{y} \\ \tau_{x y} \end{array}\right]_{k}=[\bar{Q}]_{k}\left[\begin{array}{c} \varepsilon_{x} \\ \varepsilon_{y} \\ \gamma_{x y} \end{array}\right]$$

That is, the state of tension will be continuous in the layer but discontinuous at the interface and at most it will vary linearly along the thickness due to the linear dependence of the deformation fig. 22.

### Stress

A certain fixed axial load is applied while the transverse load scaled by a multiplier, eq. (24), so as to be able to analyze the variation of internal stresses as the transverse load varies.

$$\lambda F_{\perp} \quad \operatorname{con} \lambda \in\{1 ; 2 ; 5 ; 10 ; 20 ; 30\}$$

It is evident that the displacement is proportional to the applied load, as the load multiplier increases, the deflection increases. There are no torsional effects being a symmetrical and balanced laminate.

There are no extension effects and the shift along y of the plate edge is less than 5 · 10–4. Going to see the tensions along the thickness, or rather when the coordinate is launched

\begin{equation*}
z \in\left[-\frac{h}{2} ; \frac{h}{2}\right]
\end{equation*}

It is possible to see the linear trend at times. The linear trend is given by the deformations that multiply a different stiffness coefficient layer by layer (see fig. 22). The stress also increase, in modulus, in proportion to the load multiplier.

Fig. 19. Representation of the layout considered in section §5

## Conclusions

Knowing the response of structural elements can be useful for correctly modeling a given composite laminate structure and for carrying out detailed damage studies by analyzing internal stresses.

Some cases can be as simple to predict as the alignment of all the fibers along the same direction. More complex cases, such as unbalanced layouts, can lead to structural behavior that is not always easy to predict.

Implementing an analysis of the laminated composite that takes into account the configuration of the individual layers provides useful results for understanding the structural behavior. In addition, it also allows you to take into account the internal stresses in the individual layers that present a dis-continuity at the layer-layer interface.

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# Bernoulli Bibliography

This file contains, in reverse chronological order, all items that were added to the bibliography in 1999. For references to the reviewing journals, see the main bibliography.
November 12, 1999:

KIM HAN SOO, KIM TAEKYUN,
[6] Remark on $p$-adic $q$-Bernoulli numbers. Algebraic number theory (Hapcheon/Saga, 1996). Adv. Stud. Contemp. Math. 1 (1999), 127-136.

COSTABILE F.,
[1] Expansions of real functions in Bernoulli polynomials and applications. Conf. Semin. Mat. Univ. Bari No. 273, (1999), 13 pp.

POPOV B. S.,
[1] Expressions of Laguerre polynomials through Bernoulli polynomials. Mat. Bilten No. 22, (1998), 15-18.

HAN G.-N.; RANDRIANARIVONY A.; ZENG J.,
[1] Un autre $q$-analogue des nombres d'Euler. Sém. Lothar. Combin. 42 (1999), Art. B42e, 22 pp. (electronic).

YOUNG P.T.,
[1] Congruences for Bernoulli, Euler, and Stirling numberbs. J. Number Theory, 78 (1999), no. 2, 204-227.

LAN YIZHONG,
[1] A limit formula for $\zeta(2k+1)$. J. Number Theory, 78 (1999), no. 2, 271-286.

CVIJOVIC D., KLINOWSKI J.,
[3] Values of the Legendre chi and Hurwitz zeta functions at rational arguments. Math. Comp. 68 (1999), no. 228, 1623-1630.

FUKUHARA S.,
[2] Generalized Dedekind symbols associated with the Eisenstein series. Proc. Amer. Math. Soc. 127 (1999), no. 9, 2561-2568.

FLAJOLET P., PRODINGER H.,
[1] On Stirling numbers for complex arguments and Hankel contours. SIAM J. Discrete Math. 12 (1999), no. 2, 155-159.

HAN GUO-NIU,
[2] Symétries trivariés sur les nombres de Genocchi. European J. Combin. 17 (1996), no. 4, 397--407.

HAN GUO-NIU, ZENG JIANG,
[1] On a $q$-sequence that generalizes the median Genocchi numbers. Ann. Sci. Math. Québec, 23 (1999), no. 1, 63-72.

September 10, 1999:

[1] Notes by the translator. (On L. F. Menabrea, "Sketch of the Analytical Engine Invented by Charles Babbage Esq."). In Richard Taylor (Ed.), Scientific Memoirs, Vol. III, Richard and John E. Taylor, London, 1843.

KIM E.E., TOOLE B.A.,
[1] Ada and the first computer. Scientific American, May 1999, 76-81.

MACLEOD R.A.,
[1] Fractional part sums and divisor functions, J. Number Theory 14 (1982), no. 2, 185-227.

GESSEL I.,
[4] Generating functions and generalized Dedekind sums, Electron. J. Combin. 4 (1997), no. 2, Research Paper 11, approx. 17 pp. ( electronic).

HALBRITTER U.,
[4] Some new reciprocity formulas for generalized Dedekind sums, Resultate Math. 8 (1985), no. 1, 21-46.

BOYD, D.W.,
[1] A $p$-adic study of the partial sums of the harmonic series, Experiment. Math. 3 (1994), no. 4, 287-302.

August 19, 1999:

DIAMOND J.,
[1] The $p$-adic log gamma function and $p$-adic Euler constants, Trans. Amer. Math. Soc. 233 (1977), 321-337.

KANEKO M.,
[3] Multiple zeta values and poly-Bernoulli numbers (Japanese), Seminar Reports of the Department of Mathematics, Tokyo Metropolitan University, 1997, 42 pp.

August 6, 1999:

HOFFMAN M. E.,
[2] Derivative polynomials for tangent and secant, Amer. Math. Monthly 102 (1995), no. 1, 23-30.

[3] Derivative polynomials, Euler polynomials, and associated integer sequences, Electron. J. Combin. 6 (1999), no. 1, Research Paper 21, 13 pp. (electron ic)

PORUBSKÝ S.,
[7] Voronoi Type Congruences for Bernoulli Numbers, in: "Voronoi's Impact on Modern Science" (P. Engel and H. Syta, eds.), Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, 1998.

RIORDAN J.,
[1] Inverse relations and combinatorial identities, Amer. Math. Monthly 71 1964, 485-498.

ZHU YUN HUA, YANG BI CHENG,
[1] An improvement of Euler's summation formula and some inequalities for sums of powers (Chinese), Acta Sci. Natur. Univ. Sunyatseni 36 (1997), no. 4, 21-26.

ZUBER M.,
[2] Propriétés $p$-adiques de polynômes classiques, Thèse, Université de Neuchatel, 1992.

[3] Suites de Honda, Ann. Math. Blaise Pascal 2 (1995), no. 1, 307-314.

July 15, 1999:

[7] Arithmetic properties of the Nörlund polynomial $B_n^{(x)}$, Discrete Math., 204 (1999), no. 1-3, 5-13.

APOSTOL T.M.,
[9] An elementary view of Euler's summation formula, Amer. Math. Monthly, 106 (1999), no. 5, 409-418.

BRENT B.,
[1] Quadratic minima and modular forms, Experiment. Math. 7 (1998), no. 3, 257-274.

CANDELPERGHER B., COPPO M.A., DELABAERE E.,
[1] La sommation de Ramanujan, Enseign. Math. (2) 43 (1997), no. 1-2, 93-132.

CHANG KU-YOUNG, KWON SOUN-HI,
[1] Class number problem for imaginary cyclic number fields, J. Number Theory 73 (1998), no. 2, 318-338.

DIETER U.,
[1] Reciprocity theorems for Dedekind sums, IX. Mathematikertreffen Zagreb-Graz (Motovun, 1995), 11-24, Grazer Math. Ber., 328, Karl-Franzens-Univ. Graz, Graz, 1996.

GOTO K.,
[1] A twisted adjoint $L$-value of an elliptic modular form, J. Number Theory 73 (1998), no. 1, 34-46.

HOLZAPFEL R.-P.,
[1] Zeta dimension formula for Picard modular cusp forms of neat natural congruence subgroups, Abh. Math. Sem. Univ. Hamburg 68 (1998), 169-192.

HUANG I-CHIAU, HUANG SU-YUN,
[1] Bernoulli numbers and polynomials via residues, J. Number Theory, 76 (1999), no. 2, 178-193.

ISHIBASHI M.,
[4] $\bold Q$-linear relations of special values of the Estermann zeta function, Acta Arith. 86 (1998), no. 3, 239-244.

[2] Power series and asymptotic series associated with the Lerch zeta-function, Proc. Japan Acad. Ser. A Math. Sci. 74 (1998), no. 10, 167-170.

KOHNEN W.,
[1] On the proportion of quadratic character twists of $L$-functions attached to cusp forms not vanishing at the central point, J. Reine Angew. Math. 508 (1999), 179-187.

KURT V.,
[1] Remarks on higher-dimensional Dedekind sums, Math. Japon. 45 (1997), no. 2, 297-301.

LOUBOUTIN ST.,
[2] Computation of relative class numbers of imaginary abelian number fields, Experiment. Math. 7 (1998), no. 4, 293-303.

SASVÁRI Z.,
[1] An elementary proof of Binet's formula for the gamma function, Amer. Math. Monthly 106 (1999), no. 2, 156-158.

SAUZET O.,
[1] Théorie d'Iwasawa des corps $p$-rationnels et $p$-birationnels, Manuscripta Math. 96 (1998), no. 3, 263-273.

SHAFAREVICH I.R.,
[2] Selected chapters of algebra (Russian), Matem. obrazovanie, 1997, no. 2, 3-33.

SHEVELEV V.S.,
[1] On an arithmetic property of permutation numbers with a given signature associated with the Morse sequence (Russian), Izv. Vyssh. Uchebn. Zaved. Sev.-Kavk. Reg. Estestv. Nauki 1996, no. 2, 20-24, 98-99.

STRICHARTZ R.S.,
[1] Estimates for sums of eigenvalues for domains in homogeneous spaces, J. Funct. Anal. 137 (1996), no. 1, 152-190.

YOSHIDA H.,
[1] On absolute CM-periods. II, Amer. J. Math. 120 (1998), no. 6, 1199-1236.

ZUBER M.,
[1] Propriétés de congruence de certaines familles classiques de polynômes, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 8, 869-872.

July 13, 1999:

AGOH T.,
[21] Stickelberger subideals related to Kummer-type congruences, Math. Slovaca 48 (1998), no. 4, 347-364.

AGOH T., SHOJI T.,
[1] Quadratic equations over finite fields and class numbers of real quadratic fields, Monatsh. Math. 125 (1998), no. 4, 279-292.

ARAKAWA T., KANEKO M.,
[1] Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J. 153 (1999), 189-209.

May 22, 1999:

LIU GUO DONG,
[2] Higher-order multivariable Euler's polynomial and higher-order multivariable Bernoulli's polynomial, Appl. Math. Mech. (English Ed.) 19 (1998), no. 9, 895-906; translated from Appl. Math. Mech. 19 (1998), no. 9, 827-836 (Chinese).

BARTZ K.,
[1] On Carlitz theorem for Bernoulli polynomials. Number theory (Cieszyn, 1998). Ann. Math. Sil. No. 12 (1998), 9-13.

SATOH J.,
[2] The Iwasawa $\lambda\sb p$-invariants of $\Gamma$-transforms of the generating functions of the Bernoulli numbers, Japan. J. Math. (N.S.) 17 (1991), no. 1, 165-174.

HAUSS M.,
[3] A Boole-type formula involving conjugate Euler polynomials. Charlemagne and his heritage. 1200 years of civilization and science in Europe, Vol. 2 (Aachen, 1995), 361-375, Brepols, Turnhout, 1998.

TUAN VU KIM; NGUYEN THI TINH,
[3] Expressions of Legendre polynomials through Euler polynomials, Math. Balkanica (N.S.) 11 (1997), no. 3-4, 295-302.

March 12, 1999:

PFISTER F.,
[1] Bernoulli numbers and rotational kinematics, Trans. ASME J. Appl. Mech. 65 (1998), no. 3, 758-763.

LIU GUO DONG,
[1] $n$-variable Euler numbers and polynomials, and $n$-variable Bernoulli numbers and polynomials. (Chinese), J. Math. (Wuhan) 17 (1997), no. 3, 352-358.

DUMONT D., ZENG J.,
[2] Polynômes d'Euler et fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998), no. 3, 387-410.

SATOH J.,
[4] Sums of products of two $q$-Bernoulli numbers, J. Number Theory, 74 (1999), 173-180

WRIGGE S.,
[1] Calculation of the Taylor series expansion coefficients of the Jacobian elliptic unction sn(x,k), Math. Comp. 37 (1981), no. 156, 495-497.

January 25, 1999:

SZENES, A.,
[1] Iterated residues and multiple Bernoulli polynomials, Internat. Math. Res. Notices 1998, no. 18, 937-956.

January 13, 1999:

JAKUBEC S.,
[8] Note on Wieferich's congruence for primes $p \equiv 1 \pmod{4}$, Abh. Math. Sem. Univ. Hamburg, 68 (1998), 193-197

SCHIKHOF W.H.,
[1] Ultrametric calculus. An introduction to $p$-adic analysis. Cambridge University Press, Cambridge-New York, 1984. viii+306 pp.

January 7, 1999:

[6] 2-adic congruences of Nörlund numbers and of Bernoulli numbers of the second kind, J. Number Theory, 73 (1998), no. 1, 47-58.

SKULA L.,
[18] Fermat and Wilson quotients for $p$-adic integers, Acta Math. Inform. Univ. Ostraviensis, 6 (1998), 167-181.

HOWARD F.T.,
[20] Lacunary recurrences for sums of powers of integers, Fibonacci Quart., 36 (1998), no. 5, 435-442

GLAISHER J.W.L.,
[39] Numerical values of the series $1-\frac{1}{3^n}+\frac{1}{5^n}- \frac{1}{7^n}+\frac{1}{9^n}-$ \&c., Mess. Math. (2), 42 (1913), 35-58.

January 2, 1999:

CHEN MING-PO, SRIVASTAVA H.M.,
[1] Some families of series representations for the Riemann $\zeta(3)$, Result. Math., 33 (1998), no.3-4, 179-197.

CRANDALL R.E., DILCHER K., POMERANCE C.,
[1] A search for Wieferich and Wilson primes, Math. Comp., 66 (1997), no. 217, 433-449.

EIE M., LAI K.F.,
[1] On Bernoulli identities and applications, Rev. Mat. Iberoamericana, 14 (1998), no. 1, 167-213.

GERTSCH A.,
[1] Nombres harmoniques généralisés, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), no. 1, 7-10.

GIRSTMAIR K.,
[9] Class number factors and distribution of residues, Abh. Math. Sem. Univ. Hamburg, 67 (1997), 65-104.

HACHIMORI Y., ICHIMURA H.,
[1] Semi-local units modulo Gauss sums, Manuscripta Math. 95 (1998), no. 3, 377-395.

[1] An explicit formula for the Fourier coefficients of Siegel-Eisenstein series of degree $3$, Nagoya Math. J. 146 (1997), 199-223.

KIM JAE MOON,
[2] Class numbers of real quadratic fields, Bull. Austral. Math. Soc., 57 (1998), no. 2, 261-274.

KOLYVAGIN V.A.,
[2] Fermat equations over cyclotomic fields, Proc. Steklov Inst. Math., 208 (1995), 146-165; translation from Tr. Mat. Inst. Steklova, 208 (1995), 163-185.

LE MAOHUA,
[1] A note on the generalized Bernoulli sequences, Ars Combin., 44 (1996), 283-286.

McINTOSH R.J.,
[3] Franel integrals of order four, J. Austral. Math. Soc. Ser. A, 60 (1996), no. 2, 192-203.

MOREE P.,
[2] Primes in arithmetic progression having a prescribed primitive root, MPI für Math., Bonn, Preprint Ser. 1998(57).

PANCHISHKIN A.A.,
[8] Non-archimedean Mellin transform and $p$-adic $L$-functions, Vietnam J. Math., 25 (1997), no.3, 179-202.

SCHOOF R.,
[1] Minus class groups of the fields of the $l$th roots of unity, Math. Comp., 67 (1998), no. 223, 1225-1245.

UENO K., NISHIZAWA M.,
[1] Multiple gamma functions and multiple $q$-gamma functions, Publ. Res. Inst. Math. Sci., 33 (1997), no. 5, 813-838.

YU JING, YU JIU-KANG,
[1] A note on a geometric analogue of Ankeny-Artin-Chowla's conjecture, Number theory (Tiruchirapalli, 1996), 101-105, Contemp. Math., 210, Amer. Math. Soc., Providence, RI, 1998.

DRYANOV D., KOUNCHEV O.,
[1] Polyharmonically exact formula of Euler-Maclaurin, multivariate Bernoulli functions, and Poisson type formula, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), no. 5, 515-520.

POLOVINKIN V. I.,
[1] Approximations of the Bernoulli polynomials by constants and applications to the theory of quadrature formulas, Siberian Adv. Math., 8 (1998), no. 2, 110-121.

format: plain


text:
Freeness of modules along ring homomorphisms

This question arises from my discussion with a Master student. It concerns with the following situation: let $\phi: R \to S$ be a homomorphism between Noetherian commutative rings. Suppose the $R$-module structure of $S$ has a presentation matrix with all entries in the Jacobson radical of $R$ (so $S$, as $R$-module, is the cokernel of such matrix). Let $M$ be a finitely generated, projective $R$-module.

Question: If $M \otimes_R S$ is $S$-free, must $M$ be $R$-free?

Remark: this is trivially true if $R$ is semi-local. It is easy and well-known if $\phi$ is surjective. Without some conditions on $S$, the assertion is false, for example if we take $S$ to be a residue field of a maximal ideal in $R$ and $M$ be some projective, non-free module.

This kind of statement sounds like it should be in Bourbaki or EGA or the stack project (if it is true!). Does anyone know a proof or counter example?

-
Your assumption on $S$ is satisfied in particular if $S$ is a free $R$-module. I wouldn't be surprised if you could find a counterexample in this case (e.g. with $R$ a ring of algebraic integers, and $M$ invertible?) – Laurent Moret-Bailly Mar 9 2011 at 13:47
Dear Laurent, that sounds promising! So we want $S$ to be $R$-free, $Pic(S)$ is trivial but $Pic(R)$ is not? – Hailong Dao Mar 9 2011 at 14:02
Actually, $R=\mathbb R[x,y]/(x^2+y^2-1)$, $S=R\otimes \mathbb C$ seems to work. – Hailong Dao Mar 9 2011 at 14:19
Nice and simple! – Laurent Moret-Bailly Mar 9 2011 at 14:41
Dear Laurent, thank you for the hint. In hindsight this one probably should be asked on math.stackexchange.com! – Hailong Dao Mar 9 2011 at 20:40

Let me answer and accept this in CW so that it will not be bumped periodically as not answered by the software. It was hoped that the case of $\phi$ surjective can be generalized, but as Laurent pointed out in the comments, one can not hope to get any reasonable statement.

-

So, I guess $Pic(R)$ doesn't need to be non-trivial, then? (Since all line bundles over $S^2$ are free...)

EDIT: That was supposed to be in the comments section...

-
 @Andrew, this is $S^1$. – Hailong Dao Mar 9 2011 at 17:29 Indeed. I misread, but you still have a counterexample for $R=S^2$ and $S= R \otimes \mathbb{C}$ with your module being the tangent bundle on $S^2$, no? – Andrew Parker Mar 9 2011 at 17:43 @Andrew: you are right. My comment about Pic(R) was a response to Laurent's suggestion. – Hailong Dao Mar 9 2011 at 18:20


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# Frequency Converter

## What is Frequency Converter?

Frequency converter is a calculating tool that allows you to convert a frequency value in its standard unit to its bigger and smaller units. For understanding how our frequency converter works, one needs to understand the concept of frequency and its units. Let’s have a look at it.

## What is Frequency?

In the simplest terms frequency is the number of times a thing or event occurs in a given time period. Now if we want to define it following a more scientific approach we can say that frequency is the rate per second of a vibration (in the form of a wave). The wave could be a sound wave or an electromagnetic one (radio or light wave) which can be calculated with a frequency distribution calculator.

Frequency has two types:

Resonant frequency

Relative frequency.

## What is Resonant Frequency?

It is a type of sound wave frequency which is measured by the physical parameters i.e. vibration of a vibrating object. In other words it is a natural frequency of vibrations. At its resonant frequency a subject is easy to get vibrated than at any other frequency. It is easy to calculate through a relative frequency calculator.

## What is Relative Frequency?

A relative frequency is measured in relation to the total environment. It is the fraction of times an answer occurs. For measuring the relative frequency you divide initially calculated frequency value by the total number of trials conducted. A relative frequency calculator calculate this type of frequency in a splash.

What is the Unit of Frequency? The SI unit of frequency is Hertz (Hz). One Hertz simply means once per second; 1/s. It is the number of cycles per second of any oscillating phenomenon. For every frequency there is a reciprocal time period. This time period is the time difference between two frequencies; difference of time between one event of similar kind and the very next. What are the other Units of Frequency? Standard unit of frequency is Hertz only. But since Hz represents only one cycle per second so frequency is also expressed through bigger units such as Kilohertz (KHz),

## Megahertz (MHz), and Gigahertz (GHz).

• Kilohertz refer to how many thousands of cycles occur per second.
• Megahertz refer to how many millions of cycles occur per second.
• Gigahertz refers to how many billions of cycles occur per second.
Frequency Definitions Frequency in words
Definition of Hertz (Hz) One Hertz - One cycle per Second
Definition of Kilohertz (Khz) One Kilohertz - One Thousand cycle per Second = 1,000
Definition of Megahertz (Mhz) One Megahertz - One Million cycle per Second = 1,000,000
Definition of Gigahertz (Ghz) One Gigahertz - One Billion cycle per Second = 1,000,000,000

## How to Calculate the Frequency of a Wave?

Before our frequency converter does the job of converting between smaller and bigger units of frequency, you must have a value of frequency to begin with.

Every wave has a wavelength and travels through a medium having a certain phase speed. Automatically, it means that every wave has a frequency. And that frequency is inversely proportional to the wavelength by the phase speed (speed of wave travelling through a given medium).

This means;

$$f\;=\;\frac{v}{λ}$$

Where V is the speed of wave and λ is the wavelength.

Above mentioned formula is for the frequency of a sound wave.A sound wave has a general speed of 343 m/s in air. Frequency range of sound audible for a human ear is between 20 Hz - 20,000 Hz.

Another physical wave that carries a frequency is a light wave.

In the electromagnetic spectrum there are various frequencies of light. But what is common is the fact that we assume phase speed (v) to be equal to the speed of light (c) for all electromagnetic waves. Speed of light is equal to 3*108 m/s.

This means for an electromagnetic wave (including light) formula for calculating frequency is;

$$f\;=\;\frac{v}{λ}$$

## How Does Frequency Converter Work?

As we explained above, one hertz indicates a single cycle or vibration per second. One thousand cycles per second is 1 kilohertz, or 1KHz, and 1,000,000 cycles per second is 1 megahertz, or 1 MHz.

Frequency calculator uses arithmetic for these conversions.

We are going to give you an example of the formula for converting a value of frequency in Hz to KHz.

1Hz = 0.001kHz

or

1kHz = 1000Hz

Hertz to kilohertz formula

The frequency f in kilohertz (kHz) is equal to the frequency f in hertz (Hz) divided by 1000:

$$\text{f(kHz)}\;=\;\frac{\text{f(Hz})}{1000}$$

## Example

Convert 300 hertz to kilohertz:

$$\text{f(kHz)}\;=\;\frac{300Hz}{1000}\;=\;0.5kHz$$

Suppose it is quick requirement during a practical or exam preparation and you have to follow the whole process of conversion that we mentioned above. For sure it would create hassle and definitely it would be time consuming. Instead, why won’t you use a friendly tool; that is our frequency converter. This converter will save your precious time and all the extra hard work you were going to put in for manually doing these conversions.

Let us give you another example of conversion that would consequently explain how helpful and generous our little handy frequency converter is J

1GHz = 1000MHz

or

1MHz = 0.001GHz

## Megahertz to gigahertz formula

The frequency f in gigahertz (GHz) is equal to the frequency f in megahertz (MHz) divided by 1000:

$$\text{f(GHz)}\;=\;\frac{\text{f(MHz)}}{1000}$$

## Example

Convert 7 megahertz to gigahertz:

$$\text{f(GHz)}\;=\;\frac{7MHz}{1000}\;=\;0.007GHz$$

This is the formula that you will use and that our frequency converter uses for converting a frequency value in MHz to GHz. But, the difference is that you will do it manually and frequency converter will do it in split seconds.

## How to Use Frequency Converter?

For using this tool, you just have to;


text:
# Are there any non-WEB re-implementation of TeX Core in recent years?

I am looking for re-implementations, in recent years, of the TeX core algorithm (which Knuth wrote in ca. 1982).

It would seem to me that LaTeX, XeTeX, LuaTeX all call the TeX core, which is written in the quaint Pascal-WEB. They convert the WEB program to Pascal or C by automated tools, right? There does not seem to be fresh rewriting of TeX core in more modern languages.

Related question: More Modern Reimplementations of TeX, but that is 8 years ago. If, in this decade, there is any project of new radical rewriting, I would be very willing to study the code.

1. As I understand it, LuaTeX is a partial rewriting of TeX core in CWEB, right?

2. Besides LuaTeX, is there any completely (or almost completely) re-implementation of the whole TeX core algorithm, which does not use WEB? For example, there is a guy who reportedly implemented TeX core algorithm in Clojure, but I cannot find any trace of it on GitHub. Additionally, there is a paper, which says the authors implemented TeX core algorithm in SML, but that is very long ago, and the source does not seem to have been made public.

Since I am citing new material, this is not a duplicate, I think. Of course, do correct me if any statement is wrong, or any link is mistaken.

• luatex has not had any web sources for a long time (although the cweb is fairly close to the original in parts) Sep 11, 2019 at 8:20
• I see! Do you know where is the equivalent code to the TeX core, in the LuaTeX Repo, so that I can possibly study myself? Sep 11, 2019 at 8:38
• I wonder which algorithms have to be included to 'count' here: are we talking all of TeX or just the paragraph-breaking or ...? Sep 11, 2019 at 10:23
• Doug McKenna, first at TUG 2014, and again at TUG 2019, described and demonstrated his C++ reimplementation (called JSBox), which takes care to pass the trip test. See Doug's 2014 article for further information. I haven't seen anything written yet for this year. Sep 11, 2019 at 13:52

I'm trying to collect a list of TeX implementations here. The following is what I know.

• Firstly, LuaTeX is a complete and full-featured extension of TeX, widely used today and in active development (until recently at least). It does not have any WEB code: the code was manually translated (not an automatic translation into ugly code, the way it happens in the internals of the pdfTeX/XeTeX build process) into C a long time ago. See the article LuaTEX says goodbye to Pascal (MAPS 39, EuroTeX 2009) by Taco Hoekwater for informative details. As a running example, I'll use the `scan_left_brace` example from the article: see the corresponding code in the LuaTeX sources which is currently here. (In some places I've seen mentions of CWEB but I don't know why; this is just C code with comments.)

• If Pascal is your only problem, there is Martin Ruckert's web2w project (webpage, TUGboat article), which is indeed a translation into CWEB (using an automated tool, but one that cares about the readability of the generated output). Working off this translated code, Prof. Ruckert has managed to further develop an ebook reader application (HINT) including a second version that he demonstrated at TUG 2019. The corresponding section in the typeset CWEB listing looks like this:

generated from the `.w` source where it looks like:

``````@* Basic scanning subroutines.
Let's turn now to some procedures that \TeX\ calls upon frequently to digest
certain kinds of patterns in the input. Most of these are quite simple;
some are quite elaborate. Almost all of the routines call |get_x_token|,
which can cause them to be invoked recursively.
@^stomach@>
@^recursion@>

@ The |scan_left_brace| routine is called when a left brace is supposed to be
the next non-blank token. (The term ``left brace'' means, more precisely,
a character whose catcode is |left_brace|.) \TeX\ allows \.{\\relax} to
appear before the |left_brace|.

@p void scan_left_brace(void) /*reads a mandatory |left_brace|*/
{@+@<Get the next non-blank non-relax non-call token@>;
if (cur_cmd!=left_brace)
{@+print_err("Missing { inserted");
@.Missing \{ inserted@>
help4("A left brace was mandatory here, so I've put one in.")@/
("You might want to delete and/or insert some corrections")@/
("so that I will find a matching right brace soon.")@/
("(If you're confused by all this, try typing `I}' now.)");
back_error();cur_tok=left_brace_token+'{';cur_cmd=left_brace;
cur_chr='{';incr(align_state);
}
}
``````
• If both Pascal and web/cweb-style literate programming are problematic, there is rsTeX (my favourite for reading, maybe because I had a small part to play in its becoming available online). This is a manual translation of the WEB code into (minimal) C++ (see comments at the top of the file), and the corresponding `scan_left_brace` is here.

• If you want to get more esoteric, there are a bunch of obsolete/incomplete implementations in various states of completeness and compatibility with present-day compilers. Many of them seem to derive from the manual translation that Pat Monardo made in the late 1980s/early 1990s, which was called CommonTeX. See the corresponding code section in the different repositories here (CommonTeX), here (VorTeX) (see Pehong Chen's thesis), here (cpptex), here (tex++), and a few more that don't seem to be (NTS and ExTeX are in Java, etc). See the respective descriptions on GitHub. There are also some partial re-implementations listed.

• As Barbara Beeton mentioned in a comment, there is Doug McKenna's "JSBox" (where the "JS" doesn't stand for "JavaScript" but "Johann Sebastian"), said to be a full rewrite of TeX and to pass the trip test. At TUG 2019 he demonstrated an impressive ebook / app which consists of TeX being used to typeset all the text around certain interactive figures. The source code is not (yet?) available. There was no video recording at TUG 2019, but two videos from TUG 2014 are available here and here, along with a TUGboat article.

• If you are aware of any others, please let me know.

• I created UTF-8 version of TEX using web2w. It is available at github.com/igor-liferenko/tex. It implements the features you expect from a fully usable implementation: handling of interrupts, exit codes and trailing newlines, as well as format loading, command line arguments, editor invocation, and setting the current time. May 19, 2020 at 3:06
• @IgorLiferenko That's cool, thanks! Will add it to the answer later (not at a desktop right now). May 19, 2020 at 3:26
• My implementation with UTF-8 support differs from others in that it passes the TRIP test. May 26, 2020 at 1:01


text:
# Chiral fermions as classical massless spinning particles

1 CPT - E2 Géométrie, Physique et Symétries
CPT - Centre de Physique Théorique - UMR 7332
Abstract : Semiclassical chiral fermion models with "Berry term" are studied in a symplectic framework. In the free case, the system proposed by Stephanov and Yin [1] can be obtained from Souriau's model for a relativistic massless spinning particle [2] by "enslaving the spin". The "Berry term" is identified with the classical spin two-form in [2]. The Souriau model carries a natural Poincare symmetry that we highlight, but boosts are broken by "spin enslavement". The relation between the models allows us to derive nevertheless a Poincare symmetry of an unconventional form for [1], which is shown to act differently from the natural one carried by the model [2]. Then we propose a general coupling procedure to an external electromagnetic field for any value of the gyromagnetic ratio $g$. For $g=0$ we get curious superluminal Hall-type motions, and for $g=2$ we recover spin precession. Our coupled theory is reminiscent of but different from other previously proposed ones.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-01015128
Contributor : Christian Duval <>
Submitted on : Wednesday, June 25, 2014 - 6:16:46 PM
Last modification on : Saturday, February 20, 2021 - 3:09:30 AM

### Citation

Christian Duval, P. A. Horvathy. Chiral fermions as classical massless spinning particles. Physical Review D, American Physical Society, 2015, 91 (4), pp.045013. ⟨10.1103/PhysRevD.91.045013⟩. ⟨hal-01015128⟩

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text:
Thread: 1+1 Selfridges PRP test View Single Post
2020-11-17, 22:16   #4
paulunderwood

Sep 2002
Database er0rr

1110000010102 Posts

Quote:
 Originally Posted by Dr Sardonicus This may be a silly question, but did you perhaps mean forstep(a=3,100,2 ?
No. "n" is the number tested, although I do not see how to make it an argument of a function.

Last fiddled with by paulunderwood on 2020-11-17 at 22:23


text:
# SparkFun QwiicBus Hookup Guide

Pages

## Hardware Assembly

In this section we'll go over configuring your QwiicBus EndPoint and MidPoint as how to assemble the QwiicBus circuit. Before we start connecting anything there are a couple of things to take note of.

### Ethernet Cables

Make sure your Ethernet cables are straight-through (i.e. Pin 1 on one end of the cable is connected to Pin 1 on the other end) and not crossover. Most Ethernet cables sold are straight-through so you probably have nothing to worry about here. If you're not sure what type of Ethernet cable you have, you can test for pin continuity with a digital multimeter.

Also take the voltage drop over the Ethernet cables into account. For longer chains or for circuits with many devices on multiple MidPoints, consider using one of the alternate power configurations so the voltage drop is negligible. The formula below can help you calculate the voltage drop at different lengths of CAT-6/Ethernet cable and whether or not you will need to consider one of the alternate power options:

In this formula, I is the current through the object in Amperes and R is the resistance of the wires in Ohms. Most CAT-6/Ethernet cable will have 24AWG internal wires and depending on the quality of the cable will be either copper or aluminum. Refer to a table like this one from PowerStream to determine the resistance in Ω/1000ft or Ω/km to help calculate the approximate voltage drop.

### I2C Pull-Up Resistors

Take time to note which devices you will have connected to your EndPoint and MidPoint's I2C bus side have their pull-up resistors enabled and what voltage they are tied to. Almost all Qwiic products have pull-up resistors enabled by default that pull the SDA/SCL lines connected directly to the I2C bus to 3.3V.

The PCA9615 requires pull-up resistors on the two-wire I2C bus so each EndPoint and MidPoint must have at least one pair of pull-up resistors enabled on the two-wire side. The pull-up resistors do not translate through to the differential side of the PCA9615. To help make things a bit simpler, each QwiicBus EndPoint and MidPoint come with pull-up resistors on both SDA and SCL lines tied to VDDA. Normally, VDDA is 3.3V but can be as high as 5V on the EndPoint depending on the power configuration settings.

Note: If multiple devices are connected and you have a single device with pull-up resistors, make sure the voltage they are netted to is within the operating voltage range of all other I2C devices on the bus (normally 3.3V for Qwiic devices). If needed, you can shift the voltage to safe levels with a level shifter or voltage divider. If you're not familiar with logic levels and how to shift them, take a read through this tutorial.

### Power Configuration Settings

As we mentioned in the Hardware Overview section of this guide, both QwiicBus boards feature a plethora of jumpers as well as two pairs of dedicated power PTH pins to configure how to power your QwiicBus circuit. Let's take a closer look at the three options available for powering the QwiicBus.

Important! When using alternate power configurations make sure to double check the jumpers on all EndPoints and MidPoints on your QwiicBus are set properly per the instructions below. Any QwiicBus boards or other Qwiic devices on your bus can be damaged if the circuit is assembled improperly.

#### Default - Entire System at 3.3V

In this configuration, the entire system is powered at 3.3V and the buck regulator(s) on the MidPoint(s) are unpowered. On the MidPoint(s), the Bypass jumper is CLOSED and both sides of the PSEL jumper are OPEN. 3.3V is provided either from your development board or single-board computer through the Qwiic connector or the 3.3V pin on the primary EndPoint.

Take note, in this configuration where all power is provided over the CAT-6/Ethernet cables, your total current draw will for attached devices will be limited to ~550mA with standard CAT-6/Ethernet cables due to the physical constraints of the wires inside. Anything beyond 550mA runs the risk of damaging the wires and can create a fire hazard.

The 3.3V supply voltage is then transferred from the initial EndPoint over the CAT-6/Ethernet cables in your QwiicBus system. Note that over long lengths of wire, you will see voltage drop and devices further down your QwiicBus chain may misbehave if the voltage drops below their operating range. Refer to the formula and links in the Ethernet Cables subsection above for calculating the voltage drop in your circuit.

#### Alternate 1 - VCC1 at 5V

While the default 3.3V configuration should work just fine for shorter QwiicBus chains, we found the PCA9615 functions better when powered with 5V. Powering VCC1 with 5V also allows us to also use the buck regulator on the MidPoint to source up to 3A@3.3V to devices connected to the MidPoint. Using this configuration requires adjusting both the EndPoint and MidPoint as well as multiple power supplies.

First and most importantly, the Bypass jumper on each MidPoint must be OPEN. On each MidPoint, adjust the PSEL jumper to the "1" side (see image below).

It is also strongly recommended to OPEN the 0-1 jumper on the primary / first EndPoint to isolate VCC (VDDA) from VCC1 (VDDB) but leave the 0-1 jumper on the terminating EndPoint CLOSED. Leaving the 0-1 jumper closed will send 5V to any Qwiic devices attached to your terminating / last EndPoint.

With the jumpers adjusted and your QwiicBus circuit assembled including any peripheral devices attached to your MidPoint(s), connect your 5V source to the VCC1 and GND PTH pins on the primary EndPoint. If the 0-1 jumper is opened, 3.3V should be provided over the Qwiic connectors or through the 3.3V PTH pin to power VCC (VDDA) on the primary EndPoint.

#### Alternate 2 - VCC2 at 3.8V to 36V

For this configuration, we use two separate voltages to power the EndPoints and MidPoint(s). First and most importantly, the Bypass jumper on each MidPoint must be OPEN. Adjust the PSEL jumper to the "2" position on each MidPoint.

If you plan to power VCC1 with 5V it is recommended to OPEN the 0-1 jumper on the primary EndPoint to isolate VCC from VCC1. As covered above, we recommend leaving the 0-1 jumper CLOSED on the terminating EndPoint.

After adjusting the jumpers and assembling your QwiicBus circuit including any peripheral devices attached to your MidPoint(s), connect 3.8V to 36V to the VCC2 and GND2 PTH pins on the primary EndPoint. 3.3V for the first EndPoint should be provided from the microcontroller/SBC via the Qwiic connectors or a dedicated source through the 3.3V PTH pin.

#### Alternate 3 - VCC1 at 5V and VCC2 at 6V to 36V

This advanced configuration uses multiple power supply to allow for optimal performance of the QwiicBus over very long distances with multiple MidPoints. As mentioned above, we have found the QwiicBus operates best over long distances at 5V (particularly VDDB @5V). In this configuration, VCC1 is powered with 5V and VCC2 is powered with 6V-36V. Before connecting power supplies, a few jumpers must be adjusted.

First, the Bypass jumper on each MidPoint must be OPEN. Set the PSEL jumper on all MidPoints to the "2" position. The 0-1 jumper on the primary EndPoint is OPEN and the 0-1 jumper on the terminating EndPoint is CLOSED. This isolates VDDA and VDDB on the first EndPoint so the Qwiic connectors on the first EndPoint are at 3.3V. As we mentioned before, the Qwiic connectors and 3.3V PTH pin on the terminating EndPoint will be at 5V so be careful what is connected to that EndPoint.

After adjusting jumpers on all the QwiicBus boards and assembling the rest of your circuit (including connecting any Qwiic devices to MidPoints/EndPoints), connect the 5V power supply to the VCC1 and GND PTH pins and the power supply sourcing 6V to 36V to the VCC2 and GND2 PTH pins to the primary EndPoint.

### Assembling the QwiicBus Circuit

Important! Make sure your QwiicBus circuit is not powered when adding or removing devices to your MidPoint(s) and EndPoint. Connecting and disconnecting devices while your QwiicBus is powered can cause voltage spikes and damage the PCA9615.

After taking into consideration how you intend to configure your QwiicBus circuit it's time to assemble it. Adjust the appropriate jumpers (if necessary), connect your controller (Arduino/SBC) to the primary EndPoint along with any Qwiic/I2C devices to the EndPoints and MidPoint(s) and connect the QwiicBus boards to each other with Ethernet cable plugged into the RJ-45 connectors on each board. Once the devices are all connected together, power up the circuit. The photo below shows the QwiicBus assembled in the default configuration (all boards powered with 3.3V) so your circuit may differ.

A completed QwiicBus Kit circuit using a SparkFun RedBoard Qwiic and operating at 3.3V

With everything adjusted and connected properly, that's all you need to assemble your QwiicBus circuit. Go forth and build ridiculously long, wired I2C circuits to your heart's content!


text:
# DAE Solver (G Dataflow)

Version:
Last Modified: August 12, 2017

Solves differential algebraic equations (DAEs) with initial conditions.

Solves differential algebraic equations that you define with formulas.
Solves differential algebraic equations that you define with a strictly typed VI reference.


text:
Study guide and 3 practice problems on:

## Fundamental theorem of line integrals

If $\mathbf{F}(x,y) = \nabla \phi(x,y)$ and $C$ is any curve going from $(x_0, y_0)$ to $(x_1, y_1)$: $$\int_C \mathbf{F}\cdot d\mathbf{r} = \phi(x_1, y_1) - \phi(x_0, y_0).$$
The fundamental theorem applies if the vector field $\mathbf{F}(x,y)$ is conservative. This is, when $\mathbf{F}(x,y) = \nabla \phi(x,y)$ for some $\phi$.


text:
Hands On

# HERE Fleet Telematics Route Matching API

By Shruti Kuber | 06 September 2019

### Try HERE Maps

Create a free API key to build location-aware apps and services.

Often wonder why does the route trace show that your delivery truck drove right through an apartment building? You will find out why and ways to fix it, in the tutorial below

As much as a map is useful during the drive to help you navigate, it is equally useful in giving you insights about your trip once done. Fleet management services use this trip report to not only show where the vehicle traveled, but also information like the time when it reached a certain location, the average speed, driving behaviour, etc. This information is paramount to increase the efficiency of the fleet. While the trace of the route is pretty straight-forward for open roads, the trace of a route where the road is flanked by tall buildings can look like this,

Raw trace with skewed route

This can happen due to many reasons, where multipath effect due to the surrounding tall buildings is a major contributer. The signals received by the GNSS receiver from the satellites is interrupted by the tall buildings. This signal reflects and follows an alternative path to the receiver. The receiver thus receives two signals , with the same content but at different times. This confuses the receiver and thus it can skew the positon, by a little.

Multipath effect

To adjust for this skew in position, a complex algorithm had to be written to snap the vehicle to the nearest possible point on the road. Today it is possible by simply using the HERE Fleet Telematics Route Match API This API is a REST call which reads the raw route and returns a 'cleaned up' version of the route. To use this API, you need an app_id and app_code. If you don't have credentials yet, you can get them by signing up for free at the HERE Developer Portal

We begin by getting the route waypoints in a csv/ gpx format. Make sure that the csv contains columns named 'latitude' and 'longitude'. We then convert this file into a zipped and Base64-encoded format. This can be done using simple online tools like this We do this as the route match API accepts a zipped and Base64-encoded csv/ gpx file as a parameter while calculating the corrected route. The REST call then looks like this:

GET https://rme.api.here.com/2/matchroute.json?app_id={{YOUR_APP_ID}}&app_code={{YOUR_APP_CODE}}&routemode=truck&file=bGF0aXR1ZGUsbG9uZ2l0dWRlLHRpbWVzdGFtcCxzcGVlZF9rbWgsdmVoaWNsZUlkCjQ4LjEyMTAy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Here the parameter file is has the encoded file discussed above. The parameter routemode supports arguements "car, bus, taxi, carHOV, pedestrian, truck, delivery, emergency, motorcycle, roadtrain, custom1, bdouble, bicycle, tractorTruck". The response of this GET call incliudes a JSON object with the original waypoints and linkid. What we are interested in, is the object named TracePoints which contains the latMatched and lonMatched objects which are the matched points corresponding to the given input. You can find details about other parameters in the response here. We can now use the TracePoints.latMatched and TracePoints.lonMatched to draw a polyline within them to display the route.

var xmlhttp = new XMLHttpRequest();
var points = [];

if (this.readyState == 4 && this.status == 200) {

var myObj = JSON.parse(this.responseText);
var i;
for (i=0; i < myObj.TracePoints.length; i++){
points[i] = {"lat": myObj.TracePoints[i].latMatched, "lng": myObj.TracePoints[i].lonMatched};

}
}

};

xmlhttp.open("GET", "https://ks.github.io/hosted/route_match_response.json", true);
xmlhttp.send();

var lineString = new H.geo.LineString();

points.forEach(function(point) {
lineString.pushPoint(point);
});

lineString, { style: { lineWidth: 4 }}
));

map.setViewBounds(polyline.getBounds());

}

The url used in this GET request is where I have hosted the JSON response received from the routematch API. You can chose to handle it in any manner. The important part is the plotting of the polyline with the {"lat": myObj.TracePoints[i].latMatched, "lng": myObj.TracePoints[i].lonMatched}. You can plot both the routes side-by-side and see the difference for yourself.

Comparison of original route with the matched route

You can now get rid of illogical traces and get clean traces with a simple API. Happy coding!


text:
## Elementary Statistics: A Step-by-Step Approach with Formula Card 9th Edition

a. $0.61$ b. $0.63$ c. $0.08$
a. Not oil The probability is $p=1-0.39=0.61$ b. Natural gas or oil $p=0.39+0.24=0.63$ c. Nuclear $p=0.08$


text:
# Modelling water using the TIP3P flexible system [closed]

I am modelling water using a TIP3P flexible system. I am using the pair, bond and angle parameters given on the LAMMPS webpage. I am trying to run an $$NPT$$ simulation at $$298\,K$$ and $$1 \,\mathrm{atm}$$ - so I will be using real units in my LAMMPS simulation.

I have created a 30-by-30-by-30 angstrom box and I am placing my water molecules in a lattice in this box. I am imposing periodic boundary conditions on all sides of the box. I am using LJ and Coulomb interactions to model my system, but I am not sure what to use as a cut-off distance for the TIP3P simulation.

I expect to see a density of 0.982 g/cc, but I am not seeing that. Also, I expect pressure to be 1 atm, but my pressure in my dump output is huge - and is also negative sometimes.

What am I doing wrong? Any advice would be appreciated to debug my code.

variable        data_name      index    sys.data
variable        settings_name  index    sys.settings
variable        nsteps         index    10000
variable        avg_freq       index    500
variable        coords_freq    index    2000
variable        thermo_freq    index    500
variable        dump4avg       index    50
variable        Tinit      index    300
variable        T0         index    298
variable        Tf         index    298
variable        vseed1      index  8453
variable        vseed2      index  8892

#===========================================================
# GENERAL PROCEDURES
#===========================================================
units       real
dimension   3
newton      on
boundary    p p p
atom_style  full
#===========================================================
# FORCE FIELD DEFINITION
#===========================================================
pair_style     lj/cut/coul/cut 5 5
bond_style     harmonic
angle_style    harmonic
dihedral_style none
kspace_style   none
improper_style none
pair_modify     mix geometric shift yes

#===========================================================
# SETUP SIMULATIONS
#===========================================================
read_data $${data_name} #read sys.data #include$${settings_name} #read sys.settings

# SET RUN PARAMETERS
timestep    0.01        #real-units, in femtoseconds
run_style   verlet      # Velocity-Verlet integrator

# SET OUTPUTS
thermo_style    custom step temp vol density etotal pe ke enthalpy press #lammps predefined keywords
thermo_modify   format float %14.6f #format in log file
thermo $${thermo_freq} #store all of the above information every$${thermo_freq} time step. ${x} refers to the variable x # DECLARE RELEVANT OUTPUT VARIABLES variable my_step equal step #these are LAMMPS predefined words for timestep, temperature, density, potential energy, and so on variable my_temp equal temp variable my_rho equal density variable my_pe equal pe variable my_ke equal ke variable my_etot equal etotal variable my_ent equal enthalpy variable my_P equal press variable my_vol equal vol fix averages all ave/time $${dump4avg}$$(v_avg_freq/v_dump4avg)${avg_freq} v_my_temp v_my_etot v_my_pe v_my_ke v_my_ent v_my_P file thermo.avg

#write a file called thermo.avg, compute averages of
# CREATE COORDINATE DUMPS

dump crds all custom \${coords_freq} coords.lammpstrj id type x y z vx vy vz
dump_modify crds sort id

bond_coeff * 450 0.9572 #not sure why I have to do this here
angle_coeff * 55 104.52

#===========================================================
# RUN DYNAMICS
#===========================================================
velocity all create $${Tinit}$${vseed1} mom yes rot yes #zero out momentum and any rotation
fix dynamics all npt temp $${T0}$${Tf} 298 iso 1.0 1.0 100.0
run             $${nsteps} write_data$${data_name}.end pair ii
unfix dynamics
$$$$
`
• Possible duplicate regarding cutoffs: mattermodeling.stackexchange.com/questions/1859/… – Godzilla Aug 16 '20 at 6:04
• I would suggest that, yes. People usually use about 1.0-1.4 nm cutoffs with TIP3P so 0.5 is definitely outside of what's commonly used. That being said, I don't think that's your only problem, so you will have to reproduce the problem on a simple system, otherwise it will be very difficult to debug. – Godzilla Aug 16 '20 at 21:47
• Hi, I would recommend to see Carlos Vega's benchmark papers on water models and try to replicate their results. A 30**3 angstrom box seems too large for a first calculation, if this your first time trying MD simulations. I would recommend starting from standard 216-water box; then energy minimization with NVT equilibration. Then finally NPT production run. Keep in mind, density in NPT equilibrations will take a long time to converge, so you will possibly need to run several ns long simulations. – mykd Aug 17 '20 at 11:02
• About the pressure: when simulating an NPT ensemble, the average pressure is the one you fix, but the instantaneous one can take any value. The same holds true for the density. In general, you should first equilibrate your system and then do production runs that you need to average your observables on. – lr1985 Aug 18 '20 at 13:40
• @Godzilla I suggested before that we close such questions so that people don't waste time on them later: mattermodeling.meta.stackexchange.com/q/162/5. It seems my last comment here was a waste of time, now that I see that the asker has not been online since August 30th, and didn't respond to the last comment here on August 18th. If you support my idea, I suggest writing an answer there, about the desire to have "abandoned question" listed as a close reason. – Nike Dattani Sep 13 '20 at 19:26


text:
# Left curly brace in tabular environment

I have following code:

    \begin{IEEEeqnarray}{c}
y = \beta_0 + \sum\limits_{i=1}^{m}{\beta_i x_i} + \sum\limits_{i = 1}^{m} \sum\limits_{j \geq 1}^{m}{\beta_{ij} x_i x_j} + \epsilon
\label{doe_eq}
\end{IEEEeqnarray}

\begin{tabular}{lll}
where, & $m$ &= total number of input parameters,\\
& $x_{i}$ &= $i$th system input,\\
& $y$ &= system output,\\
& $\beta_{0}$ &= model constant,\\
& $\beta_{i}$ &= linear constant,\\
& $\beta_{ij}$ &= \left \{ \begin{tabular}{l}
interaction constant when $i$\neq$j$,\\
quadratic constant when $i$=$j$,\\
\end{tabular} \\
& $\epsilon$ &= model bias
\end{tabular}

With this I get following errors:

    line 553: Missing $inserted. &$\beta_{ij}$&= \left line 554: Missing$ inserted. interaction constant when $i$\neq
line 555: Missing $inserted. q line 557: Missing \right. inserted. & line 557: Missing$ inserted. &
: Unsupported document class (or package) detected,(caption) usage of the caption package is not recommended.
line 501: Underfull \hbox (badness 10000) in paragraph

I am not sure what is causing this exactly. Any idea?

Thank you.

• \left\{ is valid only in math mode, and you can't use it without a \right. I'd write just $\{$, maybe with an explicit size declaration. \neq should also be in math mode. Sep 17 '15 at 9:19
• I assume you wanted to have a cases environment with this brace? Also, I would not write $m$= total number of... but rather have a math formula and use \text for "total number...". I think you can achieve your goal with align* instead of the outer tabular. Sep 17 '15 at 9:32
• Which document class are you using? In particular are you in two-column mode? Sep 17 '15 at 9:41
• Yes, two column, IEEETRAN. Sep 17 '15 at 9:44

As I said in my comment, \left and \neq can be used only in math mode, and \left always requires a matching \right. Furthermore I wouldn't write $i$=$j$, i.e. I wouldn't exit math mode for the equal sign. I'd type your tabular as

\documentclass{article}
\usepackage{amsmath}
\begin{document}
where
\begin{tabular}[t]{ll}
$m$ &= total number of input parameters,\\
$x_{i}$ &= $i$th system input,\\
$y$ &= system output,\\
$\beta_{0}$ &= model constant,\\
$\beta_{i}$ &= linear constant,\\
$\beta_{ij}$ &= $\begin{cases} \text{interaction constant when } i\neq j,\\ \text{quadratic constant when } i = j, \end{cases}$\\
$\epsilon$ &= model bias
\end{tabular}
\end{document}

I have removed one column, putting the word "where" outside the tabular, and I have used the alignment option [t] to match the first line of the tabular with the word "where" (but that's a matter of taste...). Furthermore, I used the cases environment from amsmath for the conditional definition. The spacing around the = still doesn't look good to me, but I think the issue was to explain the errors...

EDIT: to improve the spacing you could use an align environment

\begin{align*}
m & =  \text{total number of input parameters,}\\
x_i & =  i\text{th system input,}\\
y & = \text{system output,}\\
\beta_0&=\text{model constant,}\\
\beta_{ij}&=
\begin{cases}
\text{interaction constant when }&i\neq j,\\
\end{cases},\\
\epsilon&=\text{model bias}
\end{align*}

or if you want the symbols to be left-aligned, an alignat environment (but that's really a matter of taste)

\begin{alignat*}{2}
&m &&= \text{total number of input parameters,}\\
&x_{i} &&= i\text{-th system input,}\\
&y &&= \text{system output,}\\
&\beta_{0} &&= \text{model constant,}\\
&\beta_{i} &&= \text{linear constant,}\\
&\beta_{ij} &&= \begin{cases}
\text{interaction constant when } i\neq j,\\
\text{quadratic constant when } i = j,
\end{cases}\\
&\epsilon &&= \text{model bias}
\end{alignat*}

• @campa Thank you. Your overall suggestions to the whole thing with explanations of everything helped me to learn something here. Also your method removed a warning about box full. Thank you again. Sep 17 '15 at 9:46
• @campa sorry for taking easy way here, but in align environment how do I put where and first line " m = ...." in the same line? Sep 17 '15 at 10:05
• @algoProg Well, if you use align* you could start with \text{where } m&=.... But align* switches to a displayed equation, which is usually centered, and the word "where" would not be aligned to the left. A dirty trick would be to use flalign*, which aligns its content to the left. Then you need a & just before the first line breaking. But I guess the vertical spacing wouldn't be great in that case. Sep 17 '15 at 12:57

Use align*:

where
\begin{align*}
m & = & \text{total number of input parameters},\\
x_i & = & i\text{th system input},\\
y & =& \text{system output},\\
\beta_0&=&\text{model constant},\\
\beta_{ij}&=&
\begin{cases}
\text{interaction constant when }&i\neq j,\\


text:
Describability of a set

As Wikipedia says, there are two ways to describe a particular set: intensional definition and extensional definition.

1. For a set able to be described by extensional definition, is it necessary and sufficient to say the set has to be countable? Do there exist sets that are not describable by extensional definition?
2. For a set able to be described by intensional definition, is this equivalent to existence of a predicate to specify the membership of the set?

As "there are only countably many predicates you can actually write down (or even possible at all with a countable alphabet)" (from a comment after this reply, courtesy of Arturo Magidin), are there only countable number of sets that can be described by intensional definition? Do there exist sets that are not describable by intensional definition?

3. Are there other ways for describing a set other than intensional definition and extensional definition?

4. Do there exist sets that are not describable ?

Thanks and regards!

-
It's not clear to me that either of these terms have well-defined or unique meanings in set theory. –  Qiaochu Yuan Jan 11 '11 at 23:26
@Tim: Wouldn't the set of all bachelors be a set describable by extensional definition? You just list all the people that are bachelors. –  PEV Jan 11 '11 at 23:32
@Qiaochu: Assuming you are talking about intensional definition and extensional definition. AFAIK, intensional definition and extensional definition are just ways of giving definitions. Here they are used to describe set but not define set. –  Tim Jan 11 '11 at 23:33
@Trevor: in set theory, the set of all bachelors is not a set. For example, in ZFC the only elements of sets are other sets. Like I said, I am not sure that "intensional" and "extensional" are well-defined notions in set theory. In particular, it is far from clear to me what it means to give an extensional definition of an infinite set. –  Qiaochu Yuan Jan 11 '11 at 23:33
@Trevor: extensional definition of a set is to list all elements of the set. –  Tim Jan 11 '11 at 23:34

The thing is, in common practice within set theory, we use parameters in our descriptions, so actually any set can be given intentionally, though not necessarily in an interesting way: $a=\{x\mid x\in a\}$. This is not so useless as it first appears. For example, every set in Goedel's constructible universe $L$ is definable using as parameters "simpler" sets, and this is an important feature of $L$ that allows us to analyze it in detail, being ultimately responsible for the fact that $L$ satisfies the axiom of choice.

As already pointed out, if we only use parameter-free formulas, then we can only describe intentionally countably many sets.

As for extensional descriptions, by the most strict of interpretations, only countable sets can be described this way and, really, we need some explicit way of mentioning their elements. Of course, in practice, we usually write things like $A=\{a_0,a_1,\dots\}$ and I would say this is as extensional as it gets, though of course it is an intentional definition $A=\{a_i\mid i\in{\mathbb N}\}$ in disguise, and even this only makes sense if we have a function $i\mapsto a_i$ to begin with.

Of course, we usually write sets by listing them as $A$ above, even if the sets are not countable, by considering well-orderings, i.e., we write their elements as a long sequence (indexed by some not necessarily countable ordinal). Under choice, any set can be listed that way.

I think this idea of "extensional" and "intentional" (or "by comprehension") definitions of sets is a bit of an unintended consequence of the "new math" idea of teaching set theory in schools. So, rather than talking about the axiom of extensionality, that sets are completely determined by their elements, somehow we ended up writing sets by their extensions (i.e., by listing their elements), and rather than talking about the axiom scheme of comprehension, we ended up writing sets by comprehension. Very strange. (I (still) remember being 9 or so and my teacher asking me to write the empty set intentionally. I don't remember my teacher ever explaining how to do this. That was very confusing to me. I guess at that age, writing $\phi$ was akin to saying that $\phi$ had to hold, so the idea of writing a property $\phi(x)$ that fails for all $x$ was completely foreign.)

-
Thanks! (1) When defining a set by "a={x∣x∈a}", isn't that already a kind of definition fallacy "circular definition"? i.e. define "a" using "a" before it is defined. So people still define a specific set this way? (2) "As for extensional descriptions, by the most strict of interpretations, only countable sets can be described this way" " if we have a function i↦ai to begin with", I think a set being countable means there exists an injective function from N to the set and how to construct the function is a different issue? –  Tim Jan 12 '11 at 1:00
Hi Tim. I am not sure what you mean by a fallacy here. The equality $a=\{x\mid x\in a\}$ is certainly true. As I said, it is kind of useless, in that it gives you no information about $a$, but it is not incorrect. It is not an equation with unique solution $a$, though. And there is a slight technical point here. In ZFC we define classes intentionally but allow parameters in the descriptions. The fact that any set can be written in this silly way shows that any set is a class. A proper class is a class that is not a set' for example $V$, the class of all sets (by Russell's paradox). –  Andres Caicedo Jan 12 '11 at 1:06
I think your definition of countable reversed an arrow somewhere: $A$ is countable iff there is an injection from $A$ into ${\mathbb N}$. If $A$ is countable, and infinite, then there is also an injection from ${\mathbb N}$ into $A$, and in fact a bijection, but this is a theorem, not a definition. Sets $A$ for which there is an injection from ${\mathbb N}$ into $A$ are called Dedekind infinite and, under the axiom of choice, this is the same as infinite. Countable, on the other hand, is more restrictive. It says that in a sense the set is small (even if it happens to be infinite). –  Andres Caicedo Jan 12 '11 at 1:09
Thanks! For definition fallacy, I mean en.wikipedia.org/wiki/Fallacies_of_definition , and it is logical fallacy. For circular definition, I mean en.wikipedia.org/wiki/Circular_definition . I agree with you a={x∣x∈a} is a true equality, because every set satisfies it. But I cannot imagine if it is used to define a (therefore unique) set, because that would be circular definition, a fallacy it is. Or am I wrong? Do people really use it to define a set (either a specific set or a general set)? –  Tim Jan 12 '11 at 1:24
Thanks! You are right about "countable", except that AFAIK the existence of an injection from A into N is the definition of A being countable, as cardinality of a set is defined in terms of such injective function. Am I wrong? –  Tim Jan 12 '11 at 1:29

For any reasonable definition of "describable," the answer to #4 has to be "yes." In fact, for any reasonable definition of "describable," it should be true that almost all sets (in an appropriate sense) are not describable.

Let's take our sets to be sets in ZFC, and let's say that a set in ZFC is describable if its elements are precisely the sets $x$ satisfying a predicate $p(x)$. The correspondence between describable sets and predicates is not perfect because some (possibly most) predicates describe classes, not sets, but in any case there are at most as many describable sets as predicates in ZFC. But there are only countably many such predicates, and the class of sets is so large that it is not a set (that is, it does not have a well-defined cardinality in ZFC). So the vast majority of sets are not describable by such a predicate.

-
In fact, the situation is worse than this: the vast majority of subsets of the natural numbers are not describable. More or less equivalently, the vast majority of real numbers are not describable. Maybe the clearest illustration of this basic principle in set theory itself is that at most countably many ordinals are describable. Try to describe the ordinals, in order; it is very, very hard. –  Qiaochu Yuan Jan 11 '11 at 23:46
Can I say by "describable", you mean only "intensional definition" (or equivalently "description by predicate"?) not "extensional definition"? Are only countable sets describable ty extensional definition, and can there be countably many countable sets? –  Tim Jan 11 '11 at 23:49
@Tim: Note that a set is describable if its elements (which are sets) $x$ satisfy $p(x)$. There seems to be no formalization of "intensional" or "extensional." –  PEV Jan 11 '11 at 23:53
Edited: by "Are only countable sets describable ty extensional definition, and can there be countably many countable sets?", I wanted to say "assume only countable sets describable ty extensional definition, and can there be UNcountably many countable sets? " –  Tim Jan 11 '11 at 23:57
@Tim: like I said, you still have not provided a definition of extensional definition. The one I'm thinking of has the property that every set definable by an extensional definition can also be defined by an intensional definition. Does yours? –  Qiaochu Yuan Jan 12 '11 at 0:01


text:
# Enter Key Configuration Documentation

This sample shows how to configure the Enter and Shift+Enter keys to perform actions specified in the config.enterMode and config.shiftEnterMode settings, respectively.

You can play with different settings for the Enter and Shift+Enter keys below and then add some text in the editor. See what effect these settings have by clicking the Source toolbar button and examining the source code.

When Enter is pressed:
When Shift+Enter is pressed:

Changing the Enter Mode setting to BR or DIV is not recommended. The default CKEDITOR.ENTER_P mode is fully supported by all editor features and plugins and is also the most correct one in terms of best practices for creating web content.

If you want to change it to control paragraph spacing, you should use stylesheets instead. Edit the contents.css file and set up a suitable margin value for <p> elements, for example:

p { margin: 0; }


text:
## Oliver's blood¶

In [1]:
# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'

from thinkbayes2 import Pmf, Suite

Here is another problem from MacKay’s Information Theory, Inference, and Learning Algorithms:

Two people have left traces of their own blood at the scene of a crime. A suspect, Oliver, is tested and found to have type ‘O’ blood. The blood groups of the two traces are found to be of type ‘O’ (a common type in the local population, having frequency 60%) and of type ‘AB’ (a rare type, with frequency 1%). Do these data (type ‘O’ and ‘AB’ blood were found at scene) give evidence in favour of the proposition that Oliver was one of the two people present at the crime?

MacKay suggests formulating the problem like this:

Denote the proposition ‘the suspect and one unknown person were present’ by S. The alternative, S̄, states ‘two unknown people from the population were present’.

And then he computes the conditional probabilities of the data under S and S̄.

P(D | S) = p(AB)

P(D | S̄) = 2 p(O) p(AB)

Some people are initially unsure why there is a factor of two in the second equation. One way to convince yourself that it is correct is a verbal argument: "If Oliver did not leave a blood trace at the scene, then the blood traces were left by two unknown people. If we consider these unknown people in order, the first might have left type ‘O’ blood and the second might have left type ‘AB’, or the other way around. Since there are two ways to account for the data, we have to add their probabilities."

This is correct, but with probability it is easy for errors to hide in the words. I find it useful to express the idea computationally as well.

I'll create a Pmf object with the distribution of blood types.

In [2]:
types = Pmf({'O\t': 0.6, 'AB\t':0.01, 'other\t':0.39})
types.Print()

AB	 0.01
O	 0.6
other	 0.39

Now we can compute P(D | S) = p(AB)

In [3]:
like_S = types['AB\t']

Out[3]:
0.01

Pmf provides an addition operator that computes the distribution of all pairs of outcomes:

In [4]:
pairs = types + types
pairs.Print()

AB	AB	 0.0001
AB	O	 0.006
AB	other	 0.0039000000000000003
O	AB	 0.006
O	O	 0.36
O	other	 0.23399999999999999
other	AB	 0.0039000000000000003
other	O	 0.23399999999999999
other	other	 0.1521

Reading this table, we can see more explicitly that there are two outcomes that account for the data, AB O and O AB.

So we can compute P(D | S̄):

In [5]:
like_S̄ = pairs['O\tAB\t'] + pairs['AB\tO\t']

Out[5]:
0.012

As MacKay points out, the data are more likely under S̄ than under S, so they are evidence in favor of S̄; that is, they are exculpatory.

Let's do the update, assuming that the prior is 50:50.

In [6]:
suite = Suite(['S', 'S̄'])
suite.Print()

S 0.5
S̄ 0.5

In [7]:
suite['S'] *= like_S
suite['S̄'] *= like_S̄
suite.Normalize()

Out[7]:
0.011

In light of this evidence, we are slightly more inclined to believe that Oliver is not guilty (or at least, did not leave a blood trace at the scene).

In [8]:
suite.Print()

S 0.4545454545454546
S̄ 0.5454545454545455


text:
# What Is Multiplication? (3 Key Things To Know About It)

Multiplication is important as part of a basic arithmetic skill set.  It is used in many areas, from finance to scientific research – but what does it really mean?

So, what is multiplication?  Multiplication is a shortcut for repeated addition. When you multiply two numbers A and B (factors), you are adding A to itself B times (or adding B to itself A times). A good way to visualize this is with dots, arranged into A rows and B columns. The total number of dots will be the product A*B.

Of course, there are some rules for multiplication that you should know (such as what happens when you multiply by zero, or when you multiply two negatives together).

In this article, we’ll talk about multiplication, provide a basic multiplication table, and look at some of the rules for multiplication that you should know.

Let’s get started.

## What Is Multiplication?

Multiplication is a shortcut for repeated addition.  It helps us to save time when performing calculations.

When you multiply two numbers A and B (which are called factors), you are adding A to itself B times:

• A*B = (A + A + A + … + A)  [repeated addition with B addends; that is, A appears in the sum B times]

Alternatively, you can think of it as adding B to itself A times (since multiplication is commutative – more on this later):

What Does Propagate Mean?
What Does Propagate Mean?
• A*B = (B + B + B + … + B)  [repeated addition with A addends; that is, B appears in the sum A times]

A good way to visualize the concept of multiplication is with dots, arranged into A rows and B columns. The total number of dots will be the product A*B.

For example, if you want to multiply 4*5 (which is 20), you would arrange your dots into 4 rows and 5 columns, as illustrated below:

You can also multiply 3 or more numbers together.  A good way to visualize multiplication of 3 numbers is a 3D “tower” of dots.

For the product A*B*C, the tower would have A rows, B columns, and C layers of dots.

For multiplication of 3 or more numbers, all you need to do is complete one multiplication at a time, going from left to right.

So:

• 2*3*4
• =6*4 [since 2*3 = 6]
• =24

Likewise:

• 2*3*4*5
• =6*4*5 [since 2*3 = 6]
• =24*5 [since 6*4 = 24]
• =120

If you memorize a multiplication table, you can improve your calculation speed and avoid counting on your fingers.

### Multiplication Table

Below is a multiplication table with numbers up to 12 for both factors.

If you memorize the table above, you will have a good start on learning how to do multiplication in your head.

The table might seem overwhelming at first, since there are 12*12 = 144 entries!  However, if you learn 4 entries per day, then it will only take you 36 days (just over 5 weeks) to learn them all by heart.

### What Are The Rules Of Multiplication?

There are several rules of multiplication to be aware of, including:

• Zero factor rule
• Zero product rule
• Factor of 1 rule
• Commutativity
• Associativity
• Signs of factors & products (positive & negative)
• Multiplication by powers of 10

Let’s take a closer look at each rule, starting with zero factors.

#### Zero Factor Rule

If either factor in multiplication is zero, then the product is zero.  In other words, zero times any number is zero:

• A*B = 0 if A = 0 (for any value of B)
• A*B = 0 if B = 0 (for any value of A)

Of course, this also means that 0*0 = 0.

Remember: no matter how many numbers you multiply together, if one of them is zero, then the product is zero.

So 1*2*3*4*5*6*0 is still zero.

#### Zero Product Rule

The zero product rule is sort of the “reverse” of the zero factor rule above.  If the product of multiplication is zero, then at least one of the factors is zero:

• If A*B = 0, then either A = 0 or B = 0 (or both are zero)

Remember: no matter how many numbers you multiply together, if the product is zero, then at least one of the numbers is zero.

So if a*b*c*d*e*f*g = 0, then at least one of a, b, c, d, e, f, or g is zero.

#### Factor Of 1 Rule

For a factor of 1 in multiplication, the product is the other factor.  In other words, 1 times a number is the number itself:

• 1*B = B (for any value of B)
• A*1 = A (for any value of A)

Of course, this also means that 1*1 = 1.

Remember: no matter how many 1s you multiply together, the product is 1.

So 1*1*1*1*1*1 is still 1.

#### Commutativity

Remember that multiplication is commutative.  In other words, the order of the factors does not matter.

So:

• A*B = B*A  [for any values of A and B]

Going back to the “rows and columns of dots” analogy, the way to think of this “visually” is as follows.  You will get the same number of dots (A*B) from both of these scenarios:

• A rows of B dots
• B rows of A dots

The same rule applies for three or more factors.  For instance, with three factors A, B, and C, all six of the following expressions are equivalent:

• A*B*C
• A*C*B
• B*A*C
• B*C*A
• C*A*B
• C*B*A

#### Associativity

Remember that multiplication is associative.  In other words, the grouping of the factors does not matter.

So, we can perform any multiplication first and the result will be the same:

• (A*B)*C = A*(B*C)

Visually, these two items would have the same number of dots:

• a 3D tower of C layers, with each layer having A rows and B columns
• a 3D tower of A layers, with each layer having B rows and C columns

#### Signs Of Factors & Products (Positive & Negative)

When we multiply two factors together, the product is positive if both factors have the same sign, and negative otherwise.  That is:

• If we multiply two positive numbers, the product is positive.
• If we multiply two negative numbers, the product is positive (negatives “cancel” each other out to give us a positive).
• If we multiply a positive number and a negative number, the product is negative.

The following table summarizes the scenarios:

So:

• 2*3 = 6
• 2*(-3) = -6
• (-2)*3 = -6
• (-2)*(-3) = 6

We can apply similar reasoning for a product 3 or more factors.

If there are an even number of negative factors, then the product is positive.  Otherwise, the product is negative.

So:

• 2*3*4 = 24
• 2*3*(-4) = -24
• 2*(-3)*4 = -24
• 2*(-3)*(-4) = 24
• (-2)*3*4 = -24
• (-2)*3*(-4) = 24
• (-2)*(-3)*4 = 24
• (-2)*(-3)*(-4) = -24

#### Multiplication By Powers Of 10

If you multiply a number A by a power of 10, you can simply append zeros to the end of the number A to get the product.  Specifically, when you multiply A by 10N, append N zeroes to the end of A to get the product.

For example:

• 5*10 = 50 (A = 5, N = 1, so write A with an extra zero appended at the end)
• 16*100 = 1600 (A = 16, N = 2, so write A with two extra zeroes appended at the end)
• 25*1,000 = 25,000 (A = 25, N = 3, so write A with three extra zeroes appended at the end)
• 3*10,000 = 30,00 (A = 3, N = 4, so write A with four extra zeroes appended at the end)

## Conclusion

Now you know what multiplication is and what it means.  You also know how to visualize it, along with some of the important rules to remember.

There are lots of applications of multiplication in math, such as finding the area of a square.