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# American Institute of Mathematical Sciences

February  2012, 6(1): 39-63. doi: 10.3934/amc.2012.6.39

## Skew constacyclic codes over finite chain rings

 1 Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand, and, Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore 2 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371 3 Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

Received  September 2010 Revised  September 2011 Published  January 2012

Skew polynomial rings over finite fields and over Galois rings have recently been used to study codes. In this paper, we extend this concept to finite chain rings. Properties of skew constacyclic codes generated by monic right divisors of $x^n-\lambda$, where $\lambda$ is a unit element, are exhibited. When $\lambda^2=1$, the generators of Euclidean and Hermitian dual codes of such codes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Specializing to codes over the ring $\mathbb F$pm$+u\mathbb F$pm, the structure of all skew constacyclic codes is completely determined. This allows us to express the generators of Euclidean and Hermitian dual codes of skew cyclic and skew negacyclic codes in terms of the generators of the original codes. An illustration of all skew cyclic codes of length $2$ over $\mathbb F_3 + u\mathbb F_3$ and their Euclidean and Hermitian dual codes is also provided.
Citation: Somphong Jitman, San Ling, Patanee Udomkavanich. Skew constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2012, 6 (1) : 39-63. doi: 10.3934/amc.2012.6.39
##### References:
 [1] Y. Alkhamees, The group of automorphisms of finite chain rings,, Arab Gulf J. Sci. Res., 8 (1990), 17. Google Scholar [2] Y. Alkhamees, The determination of the group of automorphisms of a finite chain ring of characteristic $p$,, Q. J. Math., 42 (1991), 387. doi: 10.1093/qmath/42.1.387. Google Scholar [3] M. C. V. Amarra and F. R. Nemenzo, On $(1-u)$-cyclic codes over $\mathbb F$pk $+ u\mathbb F$pk,, Appl. Math. Letters, 21 (2008), 1129. doi: 10.1016/j.aml.2007.07.035. Google Scholar [4] C. Bachoc, Application of coding theory to the construction of modular lattices,, J. Combin. Theory Ser. A, 78 (1997), 92. doi: 10.1006/jcta.1996.2763. Google Scholar [5] G. Bini and F. Flamini, "Finite Commutative Rings and their Applications,", Kluwer Academic Publishers, (2002). Google Scholar [6] A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over $\mathbb F_2 + u\mathbb F_2$,, IEEE Trans. Inform. Theory, 45 (1999), 1250. doi: 10.1109/18.761278. Google Scholar [7] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes,, Appl. Algebra Engin. Commun. Comput., 18 (2007), 379. doi: 10.1007/s00200-007-0043-z. Google Scholar [8] D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings,, Adv. Math. Commun., 2 (2008), 273. doi: 10.3934/amc.2008.2.273. Google Scholar [9] D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings,, Lecture Notes Comput. Sci., 5921 (2009), 38. doi: 10.1007/978-3-642-10868-6_3. Google Scholar [10] D. Boucher and F. Ulmer, Coding with skew polynomial rings,, J. Symbol. Comput., 44 (2009), 1644. doi: 10.1016/j.jsc.2007.11.008. Google Scholar [11] W. E. Clark and D. A. Drake, Finite chain rings,, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 147. doi: 10.1007/BF02992827. Google Scholar [12] W. E. Clark and J. J. Liang, Enumeration of finite commutative chain rings,, J. Algebra, 27 (1973), 445. doi: 10.1016/0021-8693(73)90055-0. Google Scholar [13] H. Q. Dinh, Negacyclic codes of length $2^s$ over Galois rings,, IEEE Trans. Inform. Theory, 51 (2005), 4252. doi: 10.1109/TIT.2005.859284. Google Scholar [14] H. Q. Dinh, Constacyclic codes of length $2^s$ over Galois extension rings of $\mathbb F_2 + u\mathbb F_2$,, IEEE Trans. Inform. Theory, 55 (2009), 1730. doi: 10.1109/TIT.2009.2013015. Google Scholar [15] H. Q. Dinh, Constacyclic codes of length $p^s$ over $\mathbb F$pm $+ u\mathbb F$pm,, J. Algebra, 324 (2010), 940. doi: 10.1016/j.jalgebra.2010.05.027. Google Scholar [16] H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings,, IEEE Trans. Inform. Theory, 50 (2004), 1728. doi: 10.1109/TIT.2004.831789. Google Scholar [17] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at \url{http://www.codetables.de} (accessed on 2011-06-01)., (): 2011. Google Scholar [18] A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of Kerdock, Preparata, Goethals and related codes,, IEEE Trans. Inform. Theory, 40 (1994), 301. doi: 10.1109/18.312154. Google Scholar [19] T. Y. Lam, "Lectures on Modules and Rings,'', Springer-Verlag, (1999). Google Scholar [20] B. R. McDonald, "Finite Rings with Identity,", Marcel Dekker, (1974). Google Scholar [21] G. H. Norton and A. Sălăgean, On the structure of linear and cyclic codes over a finite chain ring,, Appl. Algebra Engin. Commun. Comput., 10 (2000), 489. doi: 10.1007/PL00012382. Google Scholar [22] J. F. Qian, L. N. Zhang and S. X. Zhu, $(1+u)$-cyclic and cyclic codes over the ring $\mathbb F_2 + u\mathbb F_2$,, Appl. Math. Letters, 19 (2006), 820. doi: 10.1016/j.aml.2005.10.011. Google Scholar [23] P. Ribenboim, Sur la localisation des anneaux non commutatifs (French),, in, (1972). Google Scholar [24] R. Sobhani and M. Esmaeili, Cyclic and negacyclic codes over the Galois ring G$R(p^2,m)$,, Discrete Appl. Math., 157 (2009), 2892. doi: 10.1016/j.dam.2009.03.001. Google Scholar [25] P. Udaya and A. Bonnecaze, Decoding of cyclic codes over $\mathbb F_2 +u\mathbb F_2$,, IEEE Trans. Inform. Theory, 45 (1999), 2148. doi: 10.1109/18.782165. Google Scholar [26] P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings,, IEEE Trans. Inform. Theory, 44 (1998), 1492. doi: 10.1109/18.681324. Google Scholar [27] Z.-X. Wan, "Lectures on Finite Fields and Galois Rings,", World Scientific, (2003). Google Scholar

show all references

##### References:
 [1] Y. Alkhamees, The group of automorphisms of finite chain rings,, Arab Gulf J. Sci. Res., 8 (1990), 17. Google Scholar [2] Y. Alkhamees, The determination of the group of automorphisms of a finite chain ring of characteristic $p$,, Q. J. Math., 42 (1991), 387. doi: 10.1093/qmath/42.1.387. Google Scholar [3] M. C. V. Amarra and F. R. Nemenzo, On $(1-u)$-cyclic codes over $\mathbb F$pk $+ u\mathbb F$pk,, Appl. Math. Letters, 21 (2008), 1129. doi: 10.1016/j.aml.2007.07.035. Google Scholar [4] C. Bachoc, Application of coding theory to the construction of modular lattices,, J. Combin. Theory Ser. A, 78 (1997), 92. doi: 10.1006/jcta.1996.2763. Google Scholar [5] G. Bini and F. Flamini, "Finite Commutative Rings and their Applications,", Kluwer Academic Publishers, (2002). Google Scholar [6] A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over $\mathbb F_2 + u\mathbb F_2$,, IEEE Trans. Inform. Theory, 45 (1999), 1250. doi: 10.1109/18.761278. Google Scholar [7] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes,, Appl. Algebra Engin. Commun. Comput., 18 (2007), 379. doi: 10.1007/s00200-007-0043-z. Google Scholar [8] D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings,, Adv. Math. Commun., 2 (2008), 273. doi: 10.3934/amc.2008.2.273. Google Scholar [9] D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings,, Lecture Notes Comput. Sci., 5921 (2009), 38. doi: 10.1007/978-3-642-10868-6_3. Google Scholar [10] D. Boucher and F. Ulmer, Coding with skew polynomial rings,, J. Symbol. Comput., 44 (2009), 1644. doi: 10.1016/j.jsc.2007.11.008. Google Scholar [11] W. E. Clark and D. A. Drake, Finite chain rings,, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 147. doi: 10.1007/BF02992827. Google Scholar [12] W. E. Clark and J. J. Liang, Enumeration of finite commutative chain rings,, J. Algebra, 27 (1973), 445. doi: 10.1016/0021-8693(73)90055-0. Google Scholar [13] H. Q. Dinh, Negacyclic codes of length $2^s$ over Galois rings,, IEEE Trans. Inform. Theory, 51 (2005), 4252. doi: 10.1109/TIT.2005.859284. Google Scholar [14] H. Q. Dinh, Constacyclic codes of length $2^s$ over Galois extension rings of $\mathbb F_2 + u\mathbb F_2$,, IEEE Trans. Inform. Theory, 55 (2009), 1730. doi: 10.1109/TIT.2009.2013015. Google Scholar [15] H. Q. Dinh, Constacyclic codes of length $p^s$ over $\mathbb F$pm $+ u\mathbb F$pm,, J. Algebra, 324 (2010), 940. doi: 10.1016/j.jalgebra.2010.05.027. Google Scholar [16] H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings,, IEEE Trans. Inform. Theory, 50 (2004), 1728. doi: 10.1109/TIT.2004.831789. Google Scholar [17] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at \url{http://www.codetables.de} (accessed on 2011-06-01)., (): 2011. Google Scholar [18] A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of Kerdock, Preparata, Goethals and related codes,, IEEE Trans. Inform. Theory, 40 (1994), 301. doi: 10.1109/18.312154. Google Scholar [19] T. Y. Lam, "Lectures on Modules and Rings,'', Springer-Verlag, (1999). Google Scholar [20] B. R. McDonald, "Finite Rings with Identity,", Marcel Dekker, (1974). Google Scholar [21] G. H. Norton and A. Sălăgean, On the structure of linear and cyclic codes over a finite chain ring,, Appl. Algebra Engin. Commun. Comput., 10 (2000), 489. doi: 10.1007/PL00012382. Google Scholar [22] J. F. Qian, L. N. Zhang and S. X. Zhu, $(1+u)$-cyclic and cyclic codes over the ring $\mathbb F_2 + u\mathbb F_2$,, Appl. Math. Letters, 19 (2006), 820. doi: 10.1016/j.aml.2005.10.011. Google Scholar [23] P. Ribenboim, Sur la localisation des anneaux non commutatifs (French),, in, (1972). Google Scholar [24] R. Sobhani and M. Esmaeili, Cyclic and negacyclic codes over the Galois ring G$R(p^2,m)$,, Discrete Appl. Math., 157 (2009), 2892. doi: 10.1016/j.dam.2009.03.001. Google Scholar [25] P. Udaya and A. Bonnecaze, Decoding of cyclic codes over $\mathbb F_2 +u\mathbb F_2$,, IEEE Trans. Inform. Theory, 45 (1999), 2148. doi: 10.1109/18.782165. Google Scholar [26] P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings,, IEEE Trans. Inform. Theory, 44 (1998), 1492. doi: 10.1109/18.681324. Google Scholar [27] Z.-X. Wan, "Lectures on Finite Fields and Galois Rings,", World Scientific, (2003). Google Scholar
 [1] Delphine Boucher, Patrick Solé, Felix Ulmer. Skew constacyclic codes over Galois rings. Advances in Mathematics of Communications, 2008, 2 (3) : 273-292. doi: 10.3934/amc.2008.2.273 [2] Nuh Aydin, Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Esengül Saltürk. Skew constacyclic codes over the local Frobenius non-chain rings of order 16. Advances in Mathematics of Communications, 2020, 14 (1) : 53-67. doi: 10.3934/amc.2020005 [3] Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039 [4] Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79 [5] Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1 [6] Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003 [7] Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45 [8] Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018 [9] Fatmanur Gursoy, Irfan Siap, Bahattin Yildiz. Construction of skew cyclic codes over $\mathbb F_q+v\mathbb F_q$. Advances in Mathematics of Communications, 2014, 8 (3) : 313-322. doi: 10.3934/amc.2014.8.313 [10] Amit Sharma, Maheshanand Bhaintwal. A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation. Advances in Mathematics of Communications, 2018, 12 (4) : 723-739. doi: 10.3934/amc.2018043 [11] Roy Adler, Bruce Kitchens, Michael Shub. Stably ergodic skew products. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 349-350. doi: 10.3934/dcds.1996.2.349 [12] Jory Griffin, Jens Marklof. Limit theorems for skew translations. Journal of Modern Dynamics, 2014, 8 (2) : 177-189. doi: 10.3934/jmd.2014.8.177 [13] Martianus Frederic Ezerman, San Ling, Patrick Solé, Olfa Yemen. From skew-cyclic codes to asymmetric quantum codes. Advances in Mathematics of Communications, 2011, 5 (1) : 41-57. doi: 10.3934/amc.2011.5.41 [14] Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399 [15] Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889 [16] Kohei Ueno. Weighted Green functions of nondegenerate polynomial skew products on $\mathbb{C}^2$. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 985-996. doi: 10.3934/dcds.2011.31.985 [17] Kohei Ueno. Weighted Green functions of polynomial skew products on $\mathbb{C}^2$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2283-2305. doi: 10.3934/dcds.2014.34.2283 [18] Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027 [19] Jianying Fang. 5-SEEDs from the lifted Golay code of length 24 over Z4. Advances in Mathematics of Communications, 2017, 11 (1) : 259-266. doi: 10.3934/amc.2017017 [20] Roy Adler, Bruce Kitchens, Michael Shub. Errata to "Stably ergodic skew products". Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 456-456. doi: 10.3934/dcds.1999.5.456

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Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Topological Monopoles in Quantum Antiferromagnets

Version 1 : Received: 6 February 2019 / Approved: 7 February 2019 / Online: 7 February 2019 (12:59:37 CET)

How to cite: Azzouz, M. Topological Monopoles in Quantum Antiferromagnets. Preprints 2019, 2019020073 (doi: 10.20944/preprints201902.0073.v1). Azzouz, M. Topological Monopoles in Quantum Antiferromagnets. Preprints 2019, 2019020073 (doi: 10.20944/preprints201902.0073.v1).

## Abstract

While the observation of magnetic monopoles has defied all experimental attempts in high-energy physics and astrophysics, sound theoretical approaches predict they should exist, and they have indeed been observed as quasiparticle excitations in certain condensed-matter systems. This indicates that, even though they are not ubiquitous contrary to electrons, it is possible to get them as excitations above a ground state. In this report, we show that phonons or lattice shear strain generate topological monopoles in some low-dimensional quantum antiferromagnets. For the Heisenberg ladder, phonons are found to generate topological monopoles with nonzero density due to quantum spin fluctuations. For the four-leg Heisenberg tube, longitudinal shear stress generates topological monopoles with density proportional to the strain deformation. The present theory is based on mapping the spin degrees of freedom onto spinless fermions using the generalized Jordan-Wigner transformation in dimensions higher than one. The effective magnetic field generated by the motion of the spinless fermions has non-zero divergence when phonons or shear stress are present. A possible system where the present kind of monopoles could be observed is BiCu$_2$PO$_6$.

## Subject Areas

Topological monopoles; Quantum antiferromagnets; Heisenberg model; Jordan-Wigner transformation; Spin ladders; Magnetic monopoles

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# The Origin

If there is no way to go back to the static state “which is true given that we are here now in a huge universe”, what are the other possibilities for this tiny geometry to “be” in less tense state? We have four possibilities: to expand as whole; to split or divide into connected geometries, to spin or “move” or to expand into more dimensions. Maybe it will be useful now to introduce an example: think about a little girl blowing “soap” to generate bubbles from her toy. The case with Origin is so different however we want some hints to evaluate our possibilities. In the soap bubbles case: first it’s one 3D “three dimensional” bubble, then it expands as whole, then split to smaller bubbles that tend to be closer and all the time the bubbles move around themselves, each other and a virtual center. Let’s now assume the same actions order for the Origin.

The Origin will first be expanded as whole in one dimension. This will not have any sense if it’s the only thing that happened. From where did numbers, geometry and physics evolve? We will assume now the Origin introduced a second dimension “expanded in 2D”. Note that we can assume spinning or movement too. We can’t talk about speed or any other feature of this movement here. All we know is that it’s a try to go back to static state and it was not enough. What a point expanded in tow dimensions should look like? A quick answer will be a “circle”. Should this tiny circle be like normal circles? Can we use π to make our calculations? How precise should be the value of π that will be used? Dead end! Let’s be brave and search for a way to get the exact π value (if any). Another issue here: when we say 2D what coordinate system we use here? The answer is simply: we don’t know. Since we are talking about the original geometry; we are constructing the coordinate system itself. The geometry is creating a coordinate system and expands over it. The only important attribute is dimensions “independence”.

Giving exact value for π is actually a very old problem. So many criteria were used to estimate the value of π. Some of them are pure mathematical, some are pure geometrical and few use mathematical expressions with hidden “or embedded” geometrical features. What is the most promising criterion to give the ultimate value? I will propose the very old criterion using the most modern technologies and methods. That is draw and measure. We need to draw bigger circles and measure using smaller units both the circumference C and the radios r. then substitute the measured values in the circumference equation.

Let r be the circle radius, R=2r and C be the circle Circumference

$C = \pi R.$

Or

$\pi = \frac{C}{R}$

The biggest circle should touch the universe boundaries, and the smallest measuring unit is the basic unit of Matter which we are trying to describe here. Impossible! In fact no; we can still look at the universe exactly at the point when the first “Matteron” was created or evolved. We will have a good shot to calculate π exactly. But what about our Origin circle? Since we are not sure it looks like any other circle or not and at the same time the circle is our best estimation, let’s try to calculate π. To do so, let’s look at any circle diminishing to a point. What is supposed to happen just near the point state will look like our case with the Origin. Since we are seeking the value of π lets write down the following limit expression:

Limit (1): $\lim_{R\rightarrow 0} \pi = \lim_{R\rightarrow 0} \frac{C}{R}.$

For the first instance we can say that when R approach zero the limit should approach ∞. How big is ∞ anyhow? The answer is another mystery. A closer look, we can see that when R gets smaller, C also gets smaller. We have an idea about the value of π; we use the limit to get more précised value. So ∞ or a big number is not the right answer. Another good point here is why we use R instead of C to evaluate the limit? We already know that R < C hold true for all circles. So R will reach zero before C. lets now draw our circle inside a square. Since the Origin geometry is created and dominated by the circle, the square will start to collapse into an identical circle at some stage. We can evaluate the value of π just before the square stops to be a “normal” square. We can see here the square area which holds to be greater that the circle area will approach the value of the circle area beyond that stage.

Let A(Square) be the square area, A be the circle area.

$A(Square) = R^2$

$A = \frac{\pi R^2}{2}$

$\pi = \frac{2A}{R^2}$

Limit (2): $\lim_{R\rightarrow 0} \pi = 2 \lim_{R\rightarrow 0} \frac{A}{R^2}.$

Since A can be substituted by R2, it can be easily seen that π is approaching 2. What happened beyond that stage give rise to a fraction part greater than 1 that should be multiplied by 2 to evaluate π.

Lets now introduce another dimension to our spinning tiny Origin. We will have 3D expanded point. Following the same approach using a cube and a sphere we can write:

Let V be the Sphere volume

Limit (3): $\lim_{R\rightarrow 0} \pi = 3 \lim_{R\rightarrow 0} \frac{V}{R^3}.$

π value is approaching 3. Now looking back at Limit (1) we can see that π approach 1. Note that always the limit gives rise to the fraction part which is greater than 1. Note also that the value of π integer part is exactly the number of dimensions used.

Returning back to our Origin geometry, adding more dimensions is theoretically working to reduce the tense. However we know that in our Only One available universe π value is a little greater than 3. Since we eliminate any chance for fractions because we are working on the level that “numbers” are born and a fractional dimension at this level will break the dimensions independency; this measured fraction in π as we know it should have come from somewhere else.

The Origin at this stage is a tiny 3D geometry that is spinning and still not close to stability. The last action that happened is slicing or “dividing”. The origin sliced to huge number of “near” similar pieces that are all connected together. Whether you call it Big Bang or you find any other theory to calculate exactly how all those pieces come, I think it is less important. Did we ask any question about how the Origin expanded from 1 into more dimensions? This is the same thing. We don’t have any meaning for “Time” just before the slicing; “Time” is created by slicing. So what is “Time”? What is “distance” in the first place? There is a famous statement referred to Einstein. He said that “Time is what we can measure by a clock and Distance is what we can measure with a Ruler”. Even if we neglect the hard questions and take Einstein definitions, we need to define the “clock” and the “ruler” which is hard too. First: measurement needs static bases. Even if you think about Relativity to eliminate the need for ultimate static base, you will still need “mass” and “energy” to be well defined first. Second: since the universe is a huge interconnected network of spinning matterons, any measurement tool will be connected to the measurement subject. Those questions will be discussed later.

To conclude this section: we have defined the Matteron in a reasonable way. We proved that π is not a constant number. Rather it’s a measurement ratio that can tell us so much about the meaning of time and how big and old our universe is.


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# Itallows to generalize the Riemann hypothesis to the reals.

A proportionate operator is then constructed in a similar manner in terms of the non-trivial zeros of the Riemann zeta function, extending proportionately, mapping expectedly always to zero, which imposes a ratio of the primes to said zeta roots.

## The Riemann hypothesis is true according to the axiom.

### The famous conjecture known as Riemann' s hypothesis

Does anyone know the current progress in showing the Riemann hypothesis? I was only able to find paper of Conrey that says at least 40% of the zeros of the Riemann Zeta function lie on the critical line.

### The Riemann zeta function can also be defined in terms of by

The Riemann zeta function is an extremely important of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the . While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the ) that remain unproved to this day. The Riemann zeta function is defined over the complex plane for one complex variable, which is conventionally denoted (instead of the usual ) in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859). It is implemented in the as [].

## Proof of the Riemann Hypothesis utilizing the theory …

Chengyan Liu, (09/1999)[abstract:] "Through an equivalent condition on the Farey series set forth by Franel and Landau, we prove Riemann Hypothesis for the Riemann zeta-function and the Dirichlet L-function.

## Proof of the Riemann Hypothesis utilizing the

Huang, (purports to contain a sort of philosophical/psychological 'proof' of the RH)

The respected French economist Henri Berliocchi (who also seems to have extensive interests in homeopathy and mathematics) has brought out a book called (Economica, Paris, 2001: ISBN 2-7178342-6) in which he claimsto have disproved the Riemann Hypothesis.

## Proposed (dis)proofs of the Riemann Hypothesis

As the founders of relevant logic, Anderson and Belnap, urge in their canonical book Entailment, a ‘proof’ submitted to a mathematics journal in which the essential steps fail to provide a reason to believe the conclusion, e.g. a proof by explosion, would be rejected out of hand. Mark Colyvan (2008) illustrates the point by noting that no one has laid claim to a startlingly simple proof of the Riemann hypothesis:

### Conrey, J. B. "The Riemann Hypothesis." 50,341-353, 2003. .

In addition to his own discoveries, Dirichlet played a key role ininterpreting the work of Gauss,and was an influential teacher,mentoring famous mathematicians likeBernhard Riemann (who considered Dirichlet second only to Gaussamong living mathematicians),Leopold Kronecker and Gotthold Eisenstein.

### Weisstein, E. W. "Books about Riemann Zeta Function." .

He did very important work with prime numbers,proving that there is always a primebetween any and ,and working with the zeta function before Riemann did.

### Several analogues of the Riemann hypothesis have already been proved

Riemann’s Hypothesis: All the zeros of the zeta function have real part equal to > 1/2.
Proof: Let R stand for the Russell set, the set of all sets that are not members of themselves. It is straightforward to show that this set is both a member of itself and not a member of itself. Therefore, all the zeros of Riemann’s zeta function have real part equal to 1/2.


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29 October 2013

Coherence 4 - Space coherence

The Michelson setup is an amplitude-splitting interferometer, and hence it is well adapted for time coherence measurements since the two beams originate in the same point $$P_1 = P_2$$ and the time shift $$\tau$$ is simply given by the difference in arm length. To measure the space coherence, we need a wavefront-splitting interferometer, with the two beams originating in different points $$P_1 \neq P_2$$ but at the same time: $$\tau = 0$$. The simplest example is the Young interferometer.

28 October 2013

Research standards, reproducibility and disclosure

The story is from a year and a half ago, but I only heard about it today, via the LA Times and Slashdot. Researchers from Amgen attempted to reproduce results from fifty-three research papers and only succeded for six of them. They published a comment in Nature, arguing for higher standards in preclinical research.

26 October 2013

How to prepare a presentation - style tips

In a previous post I discussed technical tips for presentations. Here I will say a few words about the style and I will conclude with a third post about the content.

At some level, every point below derives from one general principle: style is subordinated to effectiveness. Each one of your choices should make for easier communication with the audience.

How to prepare a presentation - practical tips

There are many resources on how to give a good talk, but the information is not always well structured. In this post I summarize some practical points; I'll write about style and content in future posts.

Technical issues

The safest way of avoiding technical problems is to use your own computer for the presentation, but sometimes this is not an option. Below are some points to consider, for each situation:

21 October 2013

Coherence 3 - The Michelson interferometer

The Michelson interferometer (shown below) is an excellent setup for illustrating the concept of longitudinal coherence.  We will simplify the formalism of the previous post down to three essential ingredients:
• The beam consists of wave pulses of length l (corresponding to ξl in post 2).
• They are "split in half"; each half goes through one of the arms.
• The "twin" pulses arrive at the detector with a shift ΔL.

17 October 2013

Speak, memory

...one of those blessed libraries where old newspapers
are microfilmed, as all our memories should be.
V. Nabokov

In the New Yorker, a review of a book about Henry Gustave Molaison, the man without memory (without new memories, at least) and whose ability to remember was literally sucked out of his brain.

This topic (and most anything else concerning the memory) reminds me of Borges's Funes the Memorious. whose memory was "like a garbage disposal". Finally, too much memory is as debilitating as too little...

16 October 2013

Assembly kinetics of capsid proteins

Our paper on the self-assembly of norovirus capsids made the cover and the spotlight of the Journal of the American Chemical Society!

15 October 2013

Coherence 2 - Time coherence

Time coherence is directly linked to the spectral width. Indeed, a perfectly monochromatic signal is coherent at all times: a maximum now ensures a maximum $$N$$ periods later. Not so for a polychromatic source: two oscillations of slightly different frequency are in phase at $$t=0$$ and become out of phase $$N$$ periods later, as shown below.
 Image from [BW].
This defines the coherence time (and the longitudinal coherence length $$\xi _l$$), via:

\label{shift}
\xi _l = N \lambda = (N - \frac{1}{2})(\lambda + \Delta \lambda) \Rightarrow N \simeq \frac{\lambda}{2 \Delta \lambda}\Rightarrow \xi _l \simeq \frac{\lambda^2}{2 \Delta \lambda}

In conclusion, the larger the spectral width $$\Delta \lambda$$, the shorter the longitudinal correlation length $$\xi _l$$. Once again, the descriptions in the time and frequency domains are complementary.

In the next installment, we will look at time coherence in more detail, using the Michelson interferometer.

12 October 2013

Circumspice

I usually do not link without commenting, but there is not much I could add to this:

Tom Stoppard: Information is light.

Coherence 1 - Introduction

This is the first in a series of posts where I'll try to give a simplified account of optical coherence, based on a couple of lectures I presented at the HERCULES school a few years ago. The subject is quite complex and those looking for a more rigorous and complete text should check the references below.
• What is coherence?
Let us start by putting together the "diffuse knowledge" one might have about coherence:
-  Laser light is coherent, while that emitted by thermal sources is incoherent.
-  Coherence is related to the presence (or visibility) of interference fringes.
-  Coherence "decreases as the wavelength λ decreases". This statement needs some elaboration, but it is true that is much harder to achieve coherence in the X-ray range than in the optical one.
A first attempt at a definition would be:

"Coherence" is the extent to which a field maintains a constant phase relation over time or across space.
• Why should we care?
Coherence is a fundamental property of light (and waves in general). Whenever we describe a phenomenon related to wave propagation we need to make an (explicit or implicit) assumption about the coherence of that field. Undergrad level treatments of physical optics and X-ray crystallography generally assume perfect coherence. Knowing the limitations of these approaches is essential for a deeper understanding of the underlying phenomena. Moreover, the concept of coherence is fundamental for modern applications such as holography.
• Definition
The mutual coherence function of the wave field V(r,t) - taken as scalar for simplicity - is defined as:

\begin{split}
\Gamma _V (P_1, P_2, \tau) &= \left \langle V(P_1,t+\tau) \tilde{V}(P_2,t) \right \rangle \\
& \quad \text{with} \quad \left \langle F(t) \right \rangle = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T F(T) \, \text{d}t
\label{Gam}
\end{split}

where ~ stands for the complex conjugate; we need to integrate over long times T, since detectors are very slow compared to the radiation frequency (for the visible range), so any measurement will be averaged over a large number of periods.

Equation (1) by itself is not very eloquent, but its meaning should become more clear in the following posts, through simplifications and examples.

A first observation is that the coherence function Γ resembles an intensity. Indeed, when P1=P2 and τ=0, it is simply the average intensity at that point. We can thus renormalize by the intensity to obtain a degree of coherence γ :

\gamma _V (P_1, P_2, \tau) \equiv \frac{\Gamma _V (P_1, P_2, \tau)}{\sqrt{\Gamma _V (P_1, P_1, 0)}\sqrt{\Gamma _V (P_2, P_2, 0)}} = \frac{\Gamma _V (P_1, P_2, \tau)}{\sqrt{I(P_1)}\sqrt{I(P_2)}}
\label{degree}
The coherence function contains all the information we might want. However, it is quite cumbersome. In practice, it is often convenient to reduce it to only two parameters, a time (or longitudinal) coherence length, which is roughly the decay length of ΓV with τ, and a space (or transverse) coherence length, quantifying the loss of coherence as P2 moves away from P1. We will look at longitudinal coherence in the next post.
• References
[BW]   - M. Born and E. Wolf, Principles of Optics (7th ed.),
Cambridge Univ. Press (1999).
[MW]  - L. Mandel and E. Wolf, Optical Coherence and Quantum Optics,
Cambridge Univ. Press (1995).
[G]      - J. Goodman, Introduction to Fourier Optics (3rd ed.)
Roberts & Co Publishers (2005).
[W]    - E. Wolf, Introduction to the Theory of Coherence and Polarization of Light,
Cambridge Univ. Press (2007).


text:
...

# Batman Integration

## Background

The Monte Carlo algorithm is used all over the world for a number of things. In physics, it is often used to find integrals of poorly-behaved functions, and it works incredibly well for that! Let’s take a minute to describe how the Monte Carlo algorithm works and why it’s cool.

Firstly, let’s define what an integral is: it is a fancy way to find an area. For now, that’s all we need to know. Sure, there are plenty of people who make it way, way more complicated than that, but the simpler the definition, the simpler the algorithm. I like simple algorithms.

So, if an integral is just an area, what do we need to do to find that area? Well, use geometry! If we have a box, the area of the box is obviously its $width \times height$. If we have a triangle, the area is simply $\frac{1}{2} width \times height$. Generally, the more complicated the shape, the harder it is to find the shape’s area from geometry.

That said, we don’t really need to find the shape of the object we are integrating to find it’s area. Instead, we can take a simple object, find the area of that and then embed a more complicated object inside of it. If we can find the ratio of how filled the simple object is by the more complicated object inside of it, then we can find the area of the complicated object like so:

Of course, this will only work if the simple object is larger than the more complicated one, such that the complicated object falls perfectly inside of the larger object’s boundaries.

Let’s take the example of a circle and square:

In this case, the area of the circle is $\pi \times Radius^2$ and the area of the square is $4 * Radius^2$. This means that

From here, it’s obvious that

Or the area of the circle.

But now we have a question: How do we find that ratio for arbitrary shapes? Well, Monte Carlo’s solution to this is Random Sampling. If we take a bunch of points in the simple region with the more complicated shape inside, then

And that’s pretty cool. It means that if we take a whole bunch of points, we should be able to find the ratio and integrate any arbitrarily shaped / size object! For the example above:

And we see that the ratio is pretty darn close to $\frac{\pi}{4} = 0.785398$.

The most difficult part of the Monte Carlo algorithm is finding the appropriate evaluation function. For example: How did we know whether a random point on the plane was within the circle above? Well, we simply had to plug it into a function that checked to see whether a line drawn from the origin to the x-y location of the random point was greater or less than a provided radius for the circle. The code basically looks like:

// Function to check whether point is in circle or not
bool in_circle(pos loc, pos origin, double radius){

double x = loc.x - origin.x;
double y = loc.y - origin.y;

return true;
}
else{
return false;
}
}

That said, the true power of the Monte Carlo algorithm comes from the fact that we may now integrate somewhat arbitrary functions, for a good example of this, see the next section!

## Integrating the Batman Curve

Now here’s the thing: Monte Carlo doesn’t need to use boxes and it doesn’t need to integrate obvious shapes. Let’s talk about what you need to do to integrate non-obvious functions with one of our favorite superheroes: Batman!

To be clear: the Batman Curve was not created by me. I found it on reddit and found all the information I needed on this math stack exchange post. In addition, WolframMathWorld had a decent page with the integral on it. Because the Batman Curve is difficult to integrate (though not impossible by any means!) and it has a clear analytically-defined integral to compare our Monte Carlo integration algorithm against, it was the perfect choice to test some things out.

First things first, for this integral, we were not using a standard box as described above. Instead we were using an oval. Honestly, the objects don’t matter, but I thought it was worthwhile to point out that the area of the oval is simple $\pi \times Shortest\ distance \times Longest\ distance$, so $\pi \times Radius^2$ with the shortest and longest distances to the oval from the origin as the radii.

The batman curve was a little tricky, we had to split it into the following equations (image created by Jack in the previously mentioned math stack exchange post):

Which, when plotted provide:

Then we had to write a simple function to determine if a point inside the oval was within the Batman Curve or not. To do this, we divided the curve into the following sections:

And here is a visualization of the output:

Here is the function (Nothing special, really):

// Function to check if we are in batman function for monte_carlo
bool is_batman(pos dot, pos ori, double scale){

// Determine x and y locations
double pos_x = (dot.x - ori.x) / scale;
double pos_y = (dot.y - ori.y) / scale;
double temp_x, temp_y;

// Drawing left upper wing
if (pos_x <= -3){
temp_x = (-7 * sqrt(1-((pos_y * pos_y)/9.0)));
if (pos_x >= temp_x){
return true;
}
else{
return false;
}
}

// Drawing left shoulder
if (pos_x > -3 && pos_x <= -1){
temp_x = -pos_x;
temp_y = -(2.710523708715754 + (1.5 - 0.5 * temp_x))
+1.355261854357877
*sqrt(4.0-(abs(temp_x)-1)*(abs(temp_x)-1));
if (pos_y > temp_y){
return true;
}
else{
return false;
}
}

// Drawing exterior left ear
if (pos_x > -1 && pos_x <= -0.75){
temp_y = 9.0 + 8 * pos_x;
if (pos_y > -temp_y){
return true;
}
else{
return false;
}
}

// Drawing interior left ear
if (pos_x > -0.75 && pos_x <= -0.5){
temp_y = -3 * pos_x + 0.75;
if (pos_y > -temp_y){
return true;
}
else{
return false;
}

}

if (pos_x > -.5 && pos_x <= 0.5){
temp_y = 2.25;
if (pos_y > -temp_y){
return true;
}
else{
return false;
}

}

// Drawing interior right ear
if (pos_x > 0.5 && pos_x <= 0.75){
temp_y = 3 * pos_x + 0.75;
if (pos_y > -temp_y){
return true;
}
else{
return false;
}

}

// Drawing exterior right ear
if (pos_x > 0.75 && pos_x <= 1){
temp_y = 9.0 - 8 * pos_x;
if (pos_y > -temp_y){
return true;
}
else{
return false;
}
}

// Drawing right shoulder
if (pos_x <= 3 && pos_x > 1){
temp_y = -(2.710523708715754 + (1.5 - 0.5 * pos_x))
+1.355261854357877
*sqrt(4.0-(abs(pos_x)-1)*(abs(pos_x)-1));
if (pos_y > temp_y){
return true;
}
else{
return false;
}
}

// Drawing right upper wing
if (pos_x > 3){
temp_x = (7 * sqrt(1-((pos_y * pos_y)/9.0)));
if (pos_x <= temp_x){
return true;
}
else{
return false;
}

}
}
if (pos_y >= 0){
// Drawing bottom left wing
if (pos_x <= -4.0){
temp_x = (-7 * sqrt(1-((pos_y * pos_y)/9.0)));
if (pos_x >= temp_x){
return true;
}
else{
return false;
}
}
// Drawing bottom wing
if (pos_x > -4.0 && pos_x <= 4){
//return false;
temp_y = (abs(pos_x/2)
- 0.09137221374655434 * pos_x * pos_x - 3.0)
+ sqrt(1 - (abs(abs(pos_x)-2)-1)
* (abs(abs(pos_x)-2)-1));
temp_y *= -1;
if (pos_y < temp_y){
return true;
}
else{
return false;
}
}
// Drawing bottom right wing
if (pos_x >= 4.0){
//return true;
temp_x = (7 * sqrt(1-((pos_y * pos_y)/9.0)));
if (pos_x <= temp_x){
return true;
}
else{
return false;
}
}
}

return false;

}

The full code is available here. Note that it is a little messy and long. That’s because the we used Cairo to do the visualization, which added a bunch of extra lines to the code.

Overall, this was a pretty fun side-project!


text: An increase in groundwater storage in aquifers in arid areas improves water security. Most desalination water production around the globe involves the private sector in the form of “build, operate, and transfer” or “build, operate, and own” agreements. Take-or-pay contracts are the most dominant contracts in the desalination industry. The water utility buys a fixed volume of water from the desalination company over a fixed period of 20 to 25 years. The contract between the two parties is established prior to building the plant to help ensure a profitable investment for all stakeholders. This regularly implies a surplus supply of desalinated water during low water demand periods. Given the absolute water scarcity in arid regions, maximizing the banking of surplus water in an aquifer is considered in this paper. For this purpose, a numerical groundwater flow simulation model, called MODFLOW, and a heuristic multiobjective optimizer, namely, NSGA-II, are coupled to optimize the injection and recovery of seasonal excess desalinated seawater in an alluvium coastal aquifer in Oman. Dual wells are considered for injection and abstraction of the water. The optimal daily abstraction and injection rates are determined by defining a multiobjective optimization framework. The four objective functions considered in this study are maximizing the total volume of desalinated water recharged into the aquifer; minimizing the groundwater losses to the sea; minimizing seawater intrusion by minimizing the maximum seasonal mean drawdown; and maximizing the total benefit from the recharge and recovery of the desalinated water. Analysis of the results revealed that we would be able to use 84% of the excess produced desalinated water (i.e., 8.4 of the 10 Mm3/year) that is currently returned to the sea. The net benefit from storage and recovery ranged between $14.77 million/year and$17.80 million/year. The increasing number of desalination plants at the global level calls for an integrated approach to bank the excess desalinated water and to improve the water security of coastal cities in arid and semiarid regions.


text:
3.5 Alternating current versus direct current  (Page 4/10)

 Page 4 / 10

Phet explorations: generator

Generate electricity with a bar magnet! Discover the physics behind the phenomena by exploring magnets and how you can use them to make a bulb light.

Section summary

• Direct current (DC) is the flow of electric current in only one direction. It refers to systems where the source voltage is constant.
• The voltage source of an alternating current (AC) system puts out $V={V}_{0}\phantom{\rule{0.25em}{0ex}}\text{sin 2}\pi \text{ft}$ , where $V$ is the voltage at time $t$ , ${V}_{0}$ is the peak voltage, and $f$ is the frequency in hertz.
• In a simple circuit, $I=\text{V/R}$ and AC current is $I={I}_{0}\phantom{\rule{0.25em}{0ex}}\text{sin 2}\pi \text{ft}$ , where $I$ is the current at time $t$ , and ${I}_{0}={V}_{0}\text{/R}$ is the peak current.
• The average AC power is ${P}_{\text{ave}}=\frac{1}{2}{I}_{0}{V}_{0}$ .
• Average (rms) current ${I}_{\text{rms}}$ and average (rms) voltage ${V}_{\text{rms}}$ are ${I}_{\text{rms}}=\frac{{I}_{0}}{\sqrt{2}}$ and ${V}_{\text{rms}}=\frac{{V}_{0}}{\sqrt{2}}$ , where rms stands for root mean square.
• Thus, ${P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}}$ .
• Ohm’s law for AC is ${I}_{\text{rms}}=\frac{{V}_{\text{rms}}}{R}$ .
• Expressions for the average power of an AC circuit are ${P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}}$ , ${P}_{\text{ave}}=\frac{{V}_{\text{rms}}^{\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}2}}{R}$ , and ${P}_{\text{ave}}={I}_{\text{rms}}^{\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}2}R$ , analogous to the expressions for DC circuits.

Conceptual questions

Give an example of a use of AC power other than in the household. Similarly, give an example of a use of DC power other than that supplied by batteries.

Why do voltage, current, and power go through zero 120 times per second for 60-Hz AC electricity?

You are riding in a train, gazing into the distance through its window. As close objects streak by, you notice that the nearby fluorescent lights make dashed streaks. Explain.

Problem exercises

(a) What is the hot resistance of a 25-W light bulb that runs on 120-V AC? (b) If the bulb’s operating temperature is $\text{2700º}\text{C}$ , what is its resistance at $\text{2600º}\text{C}$ ?

Certain heavy industrial equipment uses AC power that has a peak voltage of 679 V. What is the rms voltage?

480 V

A certain circuit breaker trips when the rms current is 15.0 A. What is the corresponding peak current?

Military aircraft use 400-Hz AC power, because it is possible to design lighter-weight equipment at this higher frequency. What is the time for one complete cycle of this power?

2.50 ms

A North American tourist takes his 25.0-W, 120-V AC razor to Europe, finds a special adapter, and plugs it into 240 V AC. Assuming constant resistance, what power does the razor consume as it is ruined?

In this problem, you will verify statements made at the end of the power losses for [link] . (a) What current is needed to transmit 100 MW of power at a voltage of 25.0 kV? (b) Find the power loss in a $1\text{.}\text{00 -}\phantom{\rule{0.25em}{0ex}}\Omega$ transmission line. (c) What percent loss does this represent?

(a) 4.00 kA

(b) 16.0 MW

(c) 16.0%

A small office-building air conditioner operates on 408-V AC and consumes 50.0 kW. (a) What is its effective resistance? (b) What is the cost of running the air conditioner during a hot summer month when it is on 8.00 h per day for 30 days and electricity costs $9.00 cents\text{/kW}\cdot \text{h}$ ?

What is the peak power consumption of a 120-V AC microwave oven that draws 10.0 A?

2.40 kW

What is the peak current through a 500-W room heater that operates on 120-V AC power?

Two different electrical devices have the same power consumption, but one is meant to be operated on 120-V AC and the other on 240-V AC. (a) What is the ratio of their resistances? (b) What is the ratio of their currents? (c) Assuming its resistance is unaffected, by what factor will the power increase if a 120-V AC device is connected to 240-V AC?

(a) 4.0

(b) 0.50

(c) 4.0

Nichrome wire is used in some radiative heaters. (a) Find the resistance needed if the average power output is to be 1.00 kW utilizing 120-V AC. (b) What length of Nichrome wire, having a cross-sectional area of $5.00{\text{mm}}^{2}$ , is needed if the operating temperature is $\text{500º C}$ ? (c) What power will it draw when first switched on?

Find the time after $t=0$ when the instantaneous voltage of 60-Hz AC first reaches the following values: (a) ${V}_{0}/2$ (b) ${V}_{0}$ (c) 0.

(a) 1.39 ms

(b) 4.17 ms

(c) 8.33 ms

(a) At what two times in the first period following $t=0$ does the instantaneous voltage in 60-Hz AC equal ${V}_{\text{rms}}$ ? (b) $-{V}_{\text{rms}}$ ?

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
Got questions? Join the online conversation and get instant answers!


text:
# Unifying Divisibility

Level pending

Let $$m$$ be the number of integers $$n$$ with $$1 \leq n \leq 2013$$ such that the polynomial $$x^{2n} + 1 + (x+1)^{2n}$$ is divisible by $$x^2 + x + 1$$. Find $$m \pmod {1000}$$.

**Problem credit goes to Artofproblemsolving foundation.

×


text:
# Prove $f(x)=(x-3)^2 +4$ is a Convex function?

for convex functions: $$\lambda f(x_1)+(1-\lambda)f(x_2)\geq f(\lambda x_1 +(1-\lambda)x_2)$$ for $\lambda \in [0, 1]$ My solution looks like: $$\lambda (x_1-3)^2 +4\lambda +(1-\lambda)((x-3)^2 +4)\ge\bigl(\lambda x_1+(1-\lambda)x_2 - 3)^2+4$$ $$\lambda (x_1^2+9-6x_1)+4\lambda+(1-\lambda)(x_2^2+13-6x_2) \ge (\lambda x_1+(1-\lambda)x_2-3)^2 +4$$ $$\lambda x_1^2 +x_2^2-\lambda x_2^2-6\lambda x_1-6\lambda x_2+13 \ge \lambda^2x_1^2+x_2^2+\lambda^2x_2^2-\lambda x_2^2+9+2\lambda x_1-2\lambda^2 x_2 x_1-6x_2+6\lambda x_2-6\lambda x_1$$ $$0 \ge\lambda x_1^2 - \lambda^2x_1^2 +\lambda^2x_2^2 +2\lambda x_1+12\lambda x_2 -2\lambda^2x_2x_1-4$$

I am not sure if above solution is correct, but I could not figure out how to proceed from this point.

• The first line of your solution must read $\lambda (x_1-3)^2 +4\lambda +(1-\lambda)((x-3)^2 +4)\ge\bigl(\lambda x_1+(1-\lambda)x_2 - 3)^2+4$ or $\lambda ((x_1-3)^2 +4) +(1-\lambda)((x-3)^2 +4)\ge\bigl(\lambda x_1+(1-\lambda)x_2 - 3)^2+4$ – mfl Oct 12 '16 at 17:52
• yes it should looks like i made a mistake in first step – msj003 Oct 12 '16 at 17:54
• No problem. Correct it and write down your new work. – mfl Oct 12 '16 at 17:55
• Hint: prove that $g(x)=x^2$ is convex. Then prove that, in general, if $g(x)$ is convex then $f(x)=g(x-a) + b$ is also convex. – dxiv Oct 12 '16 at 18:01
• Sum of 2 convex functions is convex – Ahmad Bazzi Oct 12 '16 at 18:04

We need to show that $$\lambda (x_1-3)^2 +4\lambda +(1-\lambda)(x_2-3)^2 +4-4\lambda \ge\bigl(\lambda x_1+(1-\lambda)x_2 - 3)^2+4.$$ That is,

$$\lambda (x_1-3)^2 +(1-\lambda)(x_2-3)^2 \ge\bigl(\lambda x_1+(1-\lambda)x_2 - 3)^2.$$ Equivalently $$\lambda x_1^2+(1-\lambda)x_2^2-6\lambda x_1- 6(1-\lambda) x_2+9$$ $$\ge \lambda^2 x_1^2+(1-\lambda)^2 x_2^2+2\lambda(1-\lambda)x_1x_2-6\lambda x_1-6(1-\lambda)x_ 2+9$$

Simplifying, we arrive at

$$\lambda x_1^2+(1-\lambda)x_2^2+\ge \lambda^2 x_1^2+(1-\lambda)^2 x_2^2+2\lambda(1-\lambda)x_1x_2.$$ Arranging terms

$$\lambda(1-\lambda) x_1^2+\lambda(1-\lambda)x_2^2-2\lambda(1-\lambda)x_1x_2\ge 0.$$ This inequality is equivalent to

$$\lambda(1-\lambda)(x_1-x_2)^2=\lambda(1-\lambda)(x_1^2+x_2^2-2x_1x_2)\ge 0,$$ which clearly holds.

HINT

Use the second derivative

• I want to do it without derivatives but looks like i am stuck in this equation. – msj003 Oct 12 '16 at 17:49

Note that we can write

\begin{align} f(\lambda x_1+(1-\lambda) x_2)&=f(\lambda (x_1-3)+(1-\lambda )(x_2-3))\\\\ &=(\lambda (x_1-3)+(1-\lambda)(x_2-3))^2+4\\\\ &=(\lambda (x_1-3))^2+((1-\lambda)(x_2-3))^2\\\\ &+2\lambda (1-\lambda)(x_1-3)(x_2-3)+4 \tag 1 \end{align}

Furthermore, we can write

$$\lambda f(x_1)+(1-\lambda)f(x_2)=\lambda (x_1-3)^2+(1-\lambda)(x_2-3)^2+4 \tag 2$$

Subtracting $(1)$ from $(2)$ yields

\begin{align} \lambda f(x_1)+(1-\lambda)f(x_2)-f(\lambda x_1+(1-\lambda) x_2)&=\lambda (1-\lambda)(x_1-x_2)^2\\\\ &\ge 0 \end{align}

as was to be shown!

• You're welcome! My pleasure. -Mark – Mark Viola Oct 12 '16 at 18:19
• Please let me know how I can improve my answer. I really want to give you the best answer I can. -Mark – Mark Viola Oct 21 '16 at 16:26
• its a good answer but i can not upvote because i am new here, less than 15 reputation. – msj003 Oct 30 '16 at 23:07
• I believe that you can up vote answers to your own posted questions. – Mark Viola Nov 4 '16 at 21:40
• it says my upvote is recorded but it is not displayed publicly. – msj003 Nov 11 '16 at 4:18


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# Measuring dark energy properties with photometrically classified Pan-STARRS supernovae. II. Cosmological parameters.

Jones, D. O. and Scolnic, D. M. and Riess, A. G. and Rest, A. and Kirshner, R. P. and Berger, E. and Kessler, R. and Pan, Y.-C. and Foley, R. J. and Chornock, R. and Ortega, C. A. and Challis, P. J. and Burgett, W. S. and Chambers, K. C. and Draper, P. W. and Flewelling, H. and Huber, M. E. and Kaiser, N. and Kudritzki, R.-P. and Metcalfe, N. and Tonry, J. and Wainscoat, R. J. and Waters, C. and Gall, E. E. E. and Kotak, R. and McCrum, M. and Smartt, S. J. and Smith, K. W. (2018) 'Measuring dark energy properties with photometrically classified Pan-STARRS supernovae. II. Cosmological parameters.', Astrophysical journal., 857 (1). p. 51.

## Abstract

We use 1169 Pan-STARRS supernovae (SNe) and 195 low-z (z < 0.1) SNe Ia to measure cosmological parameters. Though most Pan-STARRS SNe lack spectroscopic classifications, in a previous paper we demonstrated that photometrically classified SNe can be used to infer unbiased cosmological parameters by using a Bayesian methodology that marginalizes over core-collapse (CC) SN contamination. Our sample contains nearly twice as many SNe as the largest previous SN Ia compilation. Combining SNe with cosmic microwave background (CMB) constraints from Planck, we measure the dark energy equation-of-state parameter w to be −0.989 ± 0.057 (stat+sys). If w evolves with redshift as w(a) = w 0 + w a (1 − a), we find w 0 = −0.912 ± 0.149 and w a = −0.513 ± 0.826. These results are consistent with cosmological parameters from the Joint Light-curve Analysis and the Pantheon sample. We try four different photometric classification priors for Pan-STARRS SNe and two alternate ways of modeling CC SN contamination, finding that no variant gives a w differing by more than 2% from the baseline measurement. The systematic uncertainty on w due to marginalizing over CC SN contamination, ${\sigma }_{w}^{\mathrm{CC}}=0.012$, is the third-smallest source of systematic uncertainty in this work. We find limited (1.6σ) evidence for evolution of the SN color-luminosity relation with redshift, a possible systematic that could constitute a significant uncertainty in future high-z analyses. Our data provide one of the best current constraints on w, demonstrating that samples with ~5% CC SN contamination can give competitive cosmological constraints when the contaminating distribution is marginalized over in a Bayesian framework.

Item Type: Article (VoR) Version of Record First Live Deposit - 26 April 2018 Download PDF (9020Kb) Peer-reviewed https://doi.org/10.3847/1538-4357/aab6b1 © 2018. The American Astronomical Society. All rights reserved. 26 Apr 2018 14:28 26 Apr 2018 16:49

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# Ubuntu – AMD driver does not work on kernel 3.19.0-26

atifglrxkernel

I am asking for a help or a fix for this black screen in Ubuntu 15.04.

The black screen contains the message starting version 219 and I am stuck in there for like a day already. Been trying to find a fix in Google but failed. So far I tried fixing it by going into the terminal using commands like:

sudo systemctl enable sddm.service -f

This command dint go through, because I don't have sddm in my system, so it failed. Next thing is, I found out, that my sddm was a lightdm, so I tried using

sudo systemctl enable lightdm.service -f

…and same result, but this time, it got a message, that it is not meant to be activated through systemctl. Lastly I tried this one:

systemctl start lightdm

This goes in. After putting my password, it went through, but it went back to the black screen. Tried hitting the shift key, when I rebooted it, but it didn't work.

This is an Ubuntu bug resulting in an incompatibility between kernel 3.19.0-26 and the fglrx-updates package.

You can boot with the previous 3.19.0-25 kernel using grub.

This bug has been fixed already, but the new package did not reach the stable repository yet. It is released today and will reach main server today. Local mirrors may be updated a bit later.

You can download and install the patched package for 15.04 now from HERE

To fix it now press Ctrl+Alt+F1 when you get stuck on black screen. Then log into the console and run there:

sudo apt-get remove linux-image-3.19.0-26-generic
sudo reboot

But after you see fglrx-updates package is updated, you will need to run:

sudo apt-get install linux-generic

to restore the meta package to get kernels updated.

For Ubuntu 14.04.3 run:

sudo apt-get install linux-generic-lts-vivid


text:
# Pearce Wiki

### Site Tools

notes:ieee_754-1985

# Differences

This shows you the differences between two versions of the page.

 notes:ieee_754-1985 [2013/02/20 12:59]andy notes:ieee_754-1985 [2013/02/24 00:18] (current)andy [NaN] Both sides previous revision Previous revision 2013/02/24 00:18 andy [NaN] 2013/02/24 00:12 andy [Zero] 2013/02/24 00:11 andy [Normalised Values] 2013/02/24 00:09 andy [IEEE 754-1985] 2013/02/20 16:19 andy 2013/02/20 12:59 andy 2013/02/14 15:42 andy created Next revision Previous revision 2013/02/24 00:18 andy [NaN] 2013/02/24 00:12 andy [Zero] 2013/02/24 00:11 andy [Normalised Values] 2013/02/24 00:09 andy [IEEE 754-1985] 2013/02/20 16:19 andy 2013/02/20 12:59 andy 2013/02/14 15:42 andy created Line 5: Line 5: Briefly, floating point numbers are a way to represent a wide numeric range in a limited set of bits by storing a limited number of significant digits. As the absolute size of the number increases, the absolute precision decreases. Briefly, floating point numbers are a way to represent a wide numeric range in a limited set of bits by storing a limited number of significant digits. As the absolute size of the number increases, the absolute precision decreases. - The principle is similar to exponential notation for numbers (e.g. **3.523 x 106**) except that IEEE floating point values use powers of 2 instead of 10. + The principle is similar to exponential notation for numbers (e.g. $3.523 \times 10^6$) except that IEEE floating point values use powers of 2 instead of 10. Line 45: Line 45: The most common form of IEEE floating point numbers is the normalised form --- this is where the exponent has a value in the valid range (once the bias has been subtracted from the unsigned value stored). As explained above, the significand stores only the digits after the leading **1**, which is implicit. The most common form of IEEE floating point numbers is the normalised form --- this is where the exponent has a value in the valid range (once the bias has been subtracted from the unsigned value stored). As explained above, the significand stores only the digits after the leading **1**, which is implicit. - If the standard C library is available, the ''[[man>frexp|frexp()]]'' function normalises a floating point value such that the fractional part will be in the range **0.5 ≤ x < 1.0**. Multiplying this value by **2** and reducing the exponent by **1** yields a value in the desired range **1.0 ≤ x < 2.0**. At this point the leading digit can then be discarded as the implicit leading **1** (see the [[#Significand]] section for details). + If the standard C library is available, the ''[[man>frexp|frexp()]]'' function normalises a floating point value such that the fractional part will be in the range $0.5 \le \times < 1.0$. Multiplying this value by **2** and reducing the exponent by **1** yields a value in the desired range $1.0 \le \times < 2.0$. At this point the leading digit can then be discarded as the implicit leading **1** (see the [[#Significand]] section for details). If ''frexp()'' is available then it should be used, as it will likely use the underlying hardware representation to avoid expensive loops. However, a naive implementation can quite simply mimic its functionality --- the following version demonstrates the principle, but a production version would also need to check for special values (zero, NaN, infinities) as well as catching under- and overflows: If ''frexp()'' is available then it should be used, as it will likely use the underlying hardware representation to avoid expensive loops. However, a naive implementation can quite simply mimic its functionality --- the following version demonstrates the principle, but a production version would also need to check for special values (zero, NaN, infinities) as well as catching under- and overflows: Line 90: Line 90: ^ Significand | Zero | ^ Significand | Zero | - A value of exactly zero is represented by a exponent and significand of zero. The sign bit may be set or unset and IEEE 754 has the concept of both a positive and negative zero. For standard comparisons, however, these will both compare equal with zero, so the comparison **-0.0 < 0.0** yields **false**. + A value of exactly zero is represented by a exponent and significand of zero. The sign bit may be set or unset and IEEE 754 has the concept of both a positive and negative zero. For standard comparisons, however, these will both compare equal with zero, so the comparison $-0.0 < 0.0$ yields **false**. To determine the sign of a floating point value including zero, the ''[[man>copysign|copysign()]]'' function can be used with a non-zero value, or the ''[[man>signbit|signbit()]]'' macro can be used more directly on some platforms (not available on WinCE, for example). To determine the sign of a floating point value including zero, the ''[[man>copysign|copysign()]]'' function can be used with a non-zero value, or the ''[[man>signbit|signbit()]]'' macro can be used more directly on some platforms (not available on WinCE, for example). Line 112: Line 112: Since any non-zero significand value is permitted, this allows a range of values to be specified. The sign bit may also be set or unset, which could be used to differentiate between different types of NaN. The main distinction is between a **quiet NaN** which has the most-significant bit of the significand set and a **signalling NaN** which has the MSB of the significand clear((This applies to most architectures, although on PA-RISC and MIPS the sense of this bit is inverted --- the IEEE 754-2008 standard clarified this such that the bit is set for quiet NaNs.)) (although the overall value must still be non-zero). Since any non-zero significand value is permitted, this allows a range of values to be specified. The sign bit may also be set or unset, which could be used to differentiate between different types of NaN. The main distinction is between a **quiet NaN** which has the most-significant bit of the significand set and a **signalling NaN** which has the MSB of the significand clear((This applies to most architectures, although on PA-RISC and MIPS the sense of this bit is inverted --- the IEEE 754-2008 standard clarified this such that the bit is set for quiet NaNs.)) (although the overall value must still be non-zero). + The intention of the **signalling NaN** is that this will raise some sort of exception, and then go on to yield a **quiet NaN** if a result is required. This means that each error is only raised once. However, the support for this on various platforms seems to differ. + + There are three types of operations which can produce a NaN value: + + * Operations which are provided an existing NaN value as an argument. + * Operations whose results are mathematically indeterminate - some examples are listed below: + * $0.0 / 0.0$ and $\pm\infty / \pm\infty$ + * $0.0 \times \pm\infty$ + * $\infty - \infty$ and equivalents + * Operations which yield complex results - some examples are listed below: + * $\sqrt{-n}$ + * $\log{-n}$ + * $\sin^{-1}{x}$ or $\cos^{-1}{x}$ where $x < -1$ or $x > 1$ + + ===== Limits ===== + + Values in the table below apply regardless of sign, since the sign bit is an independent quantity. Base 10 values are all rounded to one decimal place. + + + ^ Limit              ^ Single precision               ^^ Double precision                 ^^ + ^:::                 ^ Base 2            ^ Base 10     ^ Base 2             ^ Base 10      ^ + ^ Smallest denormal  | 2⁻²³ x 2⁻¹²⁶      | 1.4 x 10⁻⁴⁵ | 2⁻⁵² x 2⁻¹⁰²²      | 4.9 x 10⁻³²⁴ | + ^ Middle denormal    | 2⁻¹ x 2⁻¹²⁶       | 5.9 x 10⁻³⁹ | 2⁻1 x 2⁻¹⁰²²       | 1.1 x 10⁻³⁰⁸ | + ^ Largest denormal   | (1--2⁻²³) x 2⁻¹²⁶ | 1.2 x 10⁻³⁸ | (1--2⁻⁵²) x 2⁻¹⁰²² | 2.2 x 10⁻³⁰⁸ | + ^ Smallest normal    | 1 x 2⁻¹²⁶         | 1.2 x 10⁻³⁸ | 1 x 2⁻¹⁰²²         | 2.2 x 10⁻³⁰⁸ | + ^ Middle normal      | 1 x 2⁶³           | 9.2 x 10¹⁸  | 1 x 2⁵¹¹           | 6.7 x 10¹⁵³  | + ^ Largest normal     | (2--2⁻²³) x 2¹²⁷  | 3.4 x 10³⁸  | (2--2⁻⁵²) x 2¹⁰²3  | 1.8 x 10³⁰⁸  |


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# Connection between Entropy, Energy and Zustandsum

1. Sep 30, 2014

### Nikitin

Hi. Say you have a canonical ensemble, and its zustandssumme is $Z = \sum_j e^{- \beta E_j}$.

Then $$d \: ln(Z) = \frac{-d\beta}{Z} \sum_j e^{- \beta E_j} E_j - \frac{-\beta}{Z} \sum_j e^{- \beta E_j} dE_j$$

Further, my book says the second term is given by the work $dW$ done on the system. Why exactly is that so? What about added heat? As for the first term, it's $-<E> d \beta$.

So, going on, we get $$d(ln(Z) + \beta <E>) = \beta (dW + d<E>) = \beta dQ$$, where $dQ$ is added heat.

OK, now my book makes the jump and declares $$S= k_B (ln(Z) + \beta E)$$ and $$\beta = 1/k_B T$$

Uhm, but why is this so? Why can't just as well $S= C \cdot k_B (ln(Z) + \beta E)$ and $\beta = 1/ C \cdot k_B T$, where C is any number?

2. Sep 30, 2014

### Nikitin

On second reading I see that my post was very poorly formulated, as was short on time. I apologize. At any rate, what I am curious about is:

1) Why does $dW = \frac{- \beta}{Z} \sum_j e^{- \beta E_j} dE_j = dW$ (in the OP I had a sign typo in front of this term, btw. Sorry.)

2) How come C = 1 in $S= C \cdot k_B (ln(Z) + \beta E)$ and $\beta = 1/ C \cdot k_B T$?

3. Sep 30, 2014

### ShayanJ

I can only answer this part.
The $E_j$s are actually energy of different energy levels. They depend on the volume of the system and other things which are factors that affect the interactions the particles are involved in. From these other things, only positions of system's particles are properties of the system but in statistical mechanics, we're not considering the system in that much detail and so that's irrelevant. So the only relevant characteristic of the system which the energy of the energy levels depend on, is the system's volume. But exchange of energy by changing the volume of the system, is called work.

Last edited: Sep 30, 2014
4. Sep 30, 2014

### Nikitin

But why do you ignore the potential energy between the particles and their temperature? Surely if you added heat or did work to a system of particles their energy would increase either way. And if you applied an electric field onto a system of charged particles, their potential energy would increase in relation to each other..

5. Oct 1, 2014

### Nikitin

Guys, I found answers here: Statistical Mechanics for idiots. http://www.physics.dcu.ie/~jpm/PS302/Lect2.pdf

Just in case I confused somebody else along with me, the guy in the link above will explain everything in a clear fashion (for once!!).

6. Oct 1, 2014

### Jano L.

2) Comparing the equation of state of a gas $pV=Nk_B T$ (where $T$ is empiric temperature in Kelvins) to the equation of state that can be calculated for canonical ensemble of a point particle model of ideal gas, it turns out that there is agreement when we postulate that $\beta$ gives $\frac{1}{k_B T}$. Interpretation of $\beta$ as giving $1/(k_B T)$ is then assumed for all other physical models.

1) To connect this probabilistic theory with discrete states $j$ to thermodynamics, some additional assumptions are necessary. One possible assumption is that expected average work done on the system in quasi-static process can be expressed as

$$dW = \sum_j p_jdE_j$$

where $$p_j$$

is probability that system is in the state $j$.

Since $U = \sum_j p_j E_j$, then obviously expected average accepted heat is

$$dQ = \sum_j E_jdp_j.$$

This assumption does not seem trivial and may not be completely general, but it seems to work well in this case, enabling us to identify $k_B \sum_k -p_j \ln p_j$ with thermodynamic entropy $S$.

Some motivation for the above assumption can be found in the calculations based on Schroedinger's equation. Let work be done on the system; the Hamiltonian operator then depends on some parameter, say volume of the box $V$. How do the Born probabilities $|c_j|^2$ and the Hamiltonian eigenvalues $E_j$ defined by

$$\psi = \sum_j c_j\phi_j$$
$$\hat{H}\phi_j = E_j \phi_j$$
change when $V$ changes slowly in time? It turns out the Born probabilities $|c_j|^2$ remain constant, while the eigenvalues $E_j$ change with the parameter $V$.

Alternatively, we may assume that thermodynamic entropy is given by the formula $S=k_B\sum -p_j\ln p_j$ with the Boltzmann probabilities, derive the relation $TdS = \sum_j E_j dp_j$ and thus also $dQ = \sum_j E_jdp_j$ for quasi-static process.


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Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

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• # Artykuł - szczegóły

## Bulletin of the Polish Academy of Sciences. Mathematics

2007 | 55 | 3 | 243-259

## Inverse Sequences and Absolute Co-Extensors

EN

### Abstrakty

EN
Suppose that K is a CW-complex, X is an inverse sequence of stratifiable spaces, and X = limX. Using the concept of semi-sequence, we provide a necessary and sufficient condition for X to be an absolute co-extensor for K in terms of the inverse sequence X and without recourse to any specific properties of its limit. To say that X is an absolute co-extensor for K is the same as saying that K is an absolute extensor for X, i.e., that each map f:A → K from a closed subset A of X extends to a map F:X → K. In case K is a polyhedron $|K|_{CW}$ (the set |K| with the weak topology CW), we determine a similar characterization that takes into account the simplicial structure of K.

243-259

wydano
2007

### Twórcy

autor
• Department of Mathematics, University of Zagreb, Unska 3, P.O. Box 148, 10001 Zagreb, Croatia
autor
• Department of Mathematics, University of Oklahoma, Norman, OK 73019, U.S.A.