text:
# customized firemen use a high pressure hose

#### High-Pressure Hydraulic Hose Fittings | McMaster-Carr

These two-piece fittings screw directly onto the hose without the need for a separate adapter. When asseled, the fitting compresses against the hose, forming a strong seal. Also known as reusable fittings, they can be unscrewed from the hose end and used on a new hose. Use in high-pressure hydraulic appliions up to 5, 800 psi.

#### Fire Hose Basics

Hose designed to be used to coat fires beyond the incipient stage. Attack hose shall have a minimum design operating test pressure of 275 psi. Attack Hose-1” Booster Line: Rubber Hose: Used for Rubbish, Brush, Washdowns; Requires High Pressure (approx. 200 psi) Delivers low volume (approx. 50 gpm) Highly mobile - can be handled by 1 firefighter

#### Tools Firemen Use - Firefighting gear & equipment

The hose is one of the most essential pieces of equipment used by firefighters. The hose is used to transport the water from the fire truck to the nozzle inside the structure, where firemen spray the nozzle to put out the fire. The deck gun is a controllable, high-pressure water jet that is mounted to the top of fire trucks that shoots out

#### Solved: Firemen Are Shooting A Stream Of Water At A Burnin

Question: Firemen Are Shooting A Stream Of Water At A Burning Building Using A High-pressure Hose That Shoots Out The Water With A Speed Of 25.0 M/s As It Leaves The End Of The Hose. Once It Leaves The Hose, The Water Moves In Projectile Motion. The Firemen Adjust The Angle Of Elevationof The Hose Until The Water Takes 3.00s To Reach A Building 45.0m Away.

#### Water Supply for Fireground Operations - Hydrants, LDH

If possible, firefighters should use fire hydrants that are loed on the larger water mains in the area. The supply source can sometimes be identified by a paint color on the fire hydrant

#### Fire Hoses and Fittings Are Firefighter''s Lifeline - Fire

May 01, 2012· Dan Corner, president of All-American Hose, believes the trend toward using lower pressure on fire hoses, so firefighters at the end of the line have to grapple with less reaction force, might be somewhat counterproductive. “Our Ponn Conquest attack lines were developed when smooth bore high pressure was the name of the game,” Corner says.

#### Firemen use a high-pressure hose to shoot a stream of

Firemen use a high-pressure hose to shoot a stream of water at a burning building. The water has a speed of 25.0 m/s as it leaves the end of the hose and then exhibits projectile motion. The firemen adjust the angle of elevation ? of the hose until the water takes 3.00 s to reach a building 45.0 m away.

#### When a firefighter turns on a hose, water

When a firefighter turns on a hose, water comes out of the nozzle in one direction, and the hose moves back in the opposite direction. You can explain why by using Newton’s third law. The water is forced out of the hose. A reaction force pushes the hose backward. You can also use the

#### How high can a fire hose shoot? - Slate Magazine

Firefighters can also use truck-borne ladders to reach high places, but scaling a ladder with a heavy hose is often difficult. The general rule of thu is that ladder-and-hose setups are only

#### Module introduction - NWCG

The size and type of fire hose used by your department depends on the types of fires experienced most often and the tactics used by the department. Noncollapsible rubber hose . Noncollapsible rubber hose is more commonly referred to as booster hose, hard line, or high-pressure hose. You’ll use this rubber-covered hose for mobile attack on

#### 5 Best Home Fire Fighting Pump Kits - Firefighter Garage

In other words, the harder it is to get water from the source, the lower the hose pressure will be. So you need a pretty high maximum head to have an effective firefighting pump. Pressure. Quick Takeaway: “I look for a pump with either max. PSI100 and above or maximum head of 70 meters or above.” Firefighting pumps and hoses need high pressure.

#### Tools Firemen Use - Firefighting gear & equipment

The hose is one of the most essential pieces of equipment used by firefighters. The hose is used to transport the water from the fire truck to the nozzle inside the structure, where firemen spray the nozzle to put out the fire. The deck gun is a controllable, high-pressure water jet that is mounted to the top of fire trucks that shoots out

#### Firemen are shooting a stream of water at a burning

Firemen are shooting a stream of water at a burning building. A high-pressure hose shoots out the water with a speed of 27.0m/s as it leaves the hose nozzle. Once it leaves the hose, the water moves in projectile motion. The firemen adjust the angle of elevation of the hose until the water takes 3.00s to reach a building 35.0m away.

#### Fluid Flow Pressure Calculator - Gates Corporation

Enter your appliion’s fluid properties, hose and coupling specifiions, and our Fluid Flow Pressure Calculator will help identify your operational issues, so you can eliminate costly downtime, improve efficiency, and save money.

#### Firefighter holds hose which spraying high pressure water

Get a 7.000 second firefighter holds hose which spraying stock footage at 25fps. 4K and HD video ready for any NLE immediately. Choose from a wide range of similar scenes. Video clip id 1010069531. Download footage now!

#### When a firefighter turns on a hose, water

When a firefighter turns on a hose, water comes out of the nozzle in one direction, and the hose moves back in the opposite direction. You can explain why by using Newton’s third law. The water is forced out of the hose. A reaction force pushes the hose backward. You can also use the

#### Made-To-Order Hose Asselies | Discount Hydraulic Hose

Create a custom hydraulic hose assely made to your exact specifiions. In order to ensure your hose is made correctly, please refer to our graphic below regarding how to properly measure the "Overall Length" for your custom-made assely. Custom Made Hose Asselies are NON-RETURNABLE .

#### Gettin Salty Apparel | Firefighter Apparel and Firefighter

The first motorized apparatus of the FDNY was a 1909 Knox high-pressure hose wagon, shown here in front of Engine Company No. 72 at 22 East 12th Street (today the Cinema Village theater). This hose wagon was one of the first three firefighting vehicles to simultaneously arrive on the scene of the Triangle Shirtwaist fire on March 25, 1911.

#### Fluid Flow Pressure Calculator - Gates Corporation

Enter your appliion’s fluid properties, hose and coupling specifiions, and our Fluid Flow Pressure Calculator will help identify your operational issues, so you can eliminate costly downtime, improve efficiency, and save money.

#### Firemen use a high-pressure hose to shoot a strea…

Firemen use a high-pressure hose to shoot a stream of water at a burning building. The water has a speed of 25.0 m/s as it leaves the end of the hose and then exhibits projectile motion. The firemen adjust the angle of elevation $\alpha$ of the hose until the water takes 3.00 s to reach a building 45.0 m away.

#### Solved: Fireman Is Shooting A Stream Of Water At A Burning

Question: Fireman Is Shooting A Stream Of Water At A Burning Building Using A High-pressure Hose That Shoots Out The Water With A Speed Of 25.0 M/s As It Leaves The End Of The Hose. Once It Leaves The Hose, The Water Moves In Projectile Motion With Negligible Air Resistance. The Fireman Adjusts The Angle Of Elevation Theta I Of The Hose So The Water Takes 3.00

#### Answered: Firefighters use a high-pressure hose… | bartleby

Firefighters use a high-pressure hose to shoot a stream of water at a burning building. The water has a speed of 25.0 m/s as it leaves the end of the hose and then exhibits projectile motion. The firefighters adjust the angle of elevation a of the hose until the water takes 3.00 s to reach a building 45.0 m away.

#### Solved: Firemen Are Shooting A Stream Of Water At A Burnin

Question: Firemen Are Shooting A Stream Of Water At A Burning Building Using A High-pressure Hose That Shoots Out The Water With A Speed Of 25.0 M/s As It Leaves The End Of The Hose. Once It Leaves The Hose, The Water Moves In Projectile Motion. The Firemen Adjust The Angle Of Elevationof The Hose Until The Water Takes 3.00s To Reach A Building 45.0m Away.

#### Fireman Style Hose Nozzle High Pressure Car Wash Garden

Fireman Style Hose Nozzle High Pressure Car Wash Garden Watering Sprayer Fits All Standard Garden Hoses Features: 1. High grade materials: the nozzle is made of high-quality brass, insulated steel and soft rubber parts, which can be reliably used for many years for car details, chores, gardening and cleaning around garage, yard and industrial workplace.


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Question
In linux, how to you see all the alias commands you have created?

alias -p

NOTE: just "alias" works too, without the -p

Question
In linux, how to you see all the alias commands you have created?
?

Question
In linux, how to you see all the alias commands you have created?

alias -p

NOTE: just "alias" works too, without the -p

If you want to change selection, open document below and click on "Move attachment"

format: plain


text:
# American Institute of Mathematical Sciences

January  2014, 19(1): 55-72. doi: 10.3934/dcdsb.2014.19.55

## Mathematical modeling on helper T cells in a tumor immune system

 1 Graduate School of Science and Technology, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu-shi 432-8561, Japan 2 Department of Mathematical and Systems Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu-shi 432-8561, Japan 3 Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara-shi 252-5258, Japan

Received  December 2012 Revised  October 2013 Published  December 2013

Activation of CD$8^+$ cytotoxic T lymphocytes (CTLs) is naturally regarded as a major antitumor mechanism of the immune system. In contrast, CD$4^+$ T cells are commonly classified as helper T cells (HTCs) on the basis of their roles in providing help to the generation and maintenance of effective CD$8^+$ cytotoxic and memory T cells. In order to get a better insight on the role of HTCs in a tumor immune system, we incorporate the third population of HTCs into a previous two dimensional ordinary differential equations (ODEs) model. Further we introduce the adoptive cellular immunotherapy (ACI) as the treatment to boost the immune system to fight against tumors. Compared tumor cells (TCs) and effector cells (ECs), the recruitment of HTCs changes the dynamics of the system substantially, by the effects through particular parameters, i.e., the activation rate of ECs by HTCs, $p$ (scaled as $\rho$), and the HTCs stimulation rate by the presence of identified tumor antigens, $k_2$ (scaled as $\omega_2$). We describe the stability regions of the interior equilibria $E^*$ (no treatment case) and $E^+$ (treatment case) in the scaled $(\rho,\omega_2)$ parameter space respectively. Both $\rho$ and $\omega_2$ can destabilize $E^*$ and $E^+$ and cause Hopf bifurcations. Our results show that HTCs might play a crucial role in the long term periodic oscillation behaviors of tumor immune system interactions. They also show that TCs may be eradicated from the patient's body under the ACI treatment.
Citation: Yueping Dong, Rinko Miyazaki, Yasuhiro Takeuchi. Mathematical modeling on helper T cells in a tumor immune system. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 55-72. doi: 10.3934/dcdsb.2014.19.55
##### References:
 [1] J. A. Adam and N. Bellomo, A Survey of Models for Tumor-immune System Dynamics, Birkhauser, Boston, 1997. doi: 10.1007/978-0-8176-8119-7. [2] J. C. Arciero, T. L. Jackson and D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 39-58. [3] T. Boon and P. Van der Bruggen, Human tumor antigens recognized by T lymphocytes, J. Exp. Med., 183 (1996), 725-729. doi: 10.1084/jem.183.3.725. [4] C. Bourgeois, B. Rocha and C. Tanchot, A role for CD40 expression on CD8$^+$ T cells in the generation of CD8$^+$ T cell memory, Science, 297 (2002), 2060-2063. doi: 10.1126/science.1072615. [5] L. G. de Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res., 65 (2005), 7950-7958. [6] L. G. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations, J. Theor. Biol., 238 (2006), 841-862. doi: 10.1016/j.jtbi.2005.06.037. [7] K. E. de Visser, A. Eichten and L. M. Coussens, Paradoxical roles of the immune system during cancer development, Nat. Rev. Cancer, 6 (2006), 24-37. [8] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences, Phys. D, 208 (2005), 220-235. doi: 10.1016/j.physd.2005.06.032. [9] R. Eftimie, J. L. Bramson and D. J. D. Earn, Modeling anti-tumor Th1 and Th2 immunity in the rejection of melanoma, J. Theor. Biol., 265 (2010), 467-480. doi: 10.1016/j.jtbi.2010.04.030. [10] R. Eftimie, J. L. Bramson and D. J. D. Earn, Interactions between the immune system and cancer: A brief review of non-spatial mathematical models, Bull. Math. Biol., 73 (2011), 2-32. doi: 10.1007/s11538-010-9526-3. [11] B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718195. [12] P. Fortin and M. C. Mackey, Periodic chronic myelogenous leukaemia: spectral analysis of blood cell counts and aetiological implications, Brit. J. Haematol., 104 (1999), 336-345. doi: 10.1046/j.1365-2141.1999.01168.x. [13] M. Galach, Dynamics of the tumor-immune system competition-the effect of time delay, Int. J. Appl. Math. Comput. Sci., 13 (2003), 395-406. [14] K. Hung, R. Hayashi, A. Lafond-Walker, C. Lowenstein, D. Pardoll and H. Levitsky, The central role of CD4+ T cells in the antitumor immune response, J. Exp. Med., 188 (1998), 2357-2368. doi: 10.1084/jem.188.12.2357. [15] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252. doi: 10.1007/s002850050127. [16] D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Math. Biosci. Eng., 6 (2009), 573-583. doi: 10.3934/mbe.2009.6.573. [17] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 2 (1994), 295-321. [18] Y. P. Lai, C. J. Jeng and S. C. Chen, The roles of CD4$^+$ T cells in tumor immunity, ISRN Immunology, 2011 (2011), 1-6. doi: 10.5402/2011/497397. [19] C. Letellier, F. Denis and L. A. Aguirre, What can be learned from a chaotic cancer model? J. Theor. Biol., 322 (2013), 7-16. doi: 10.1016/j.jtbi.2013.01.003. [20] O. Lejeune, M. A. J. Chaplain and I. EI Akili, Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumours, Math. Comput. Modelling, 47 (2008), 649-662. doi: 10.1016/j.mcm.2007.02.026. [21] D. Liu, S. Ruan and D. Zhu, Bifurcation analysis in models of tumor and immune system interactions, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 151-168. doi: 10.3934/dcdsb.2009.12.151. [22] D. Liu, S. Ruan and D. Zhu, Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions, Math. Biosci. Eng., 9 (2012), 347-368. doi: 10.3934/mbe.2012.9.347. [23] M. F. Mackey, R. J. Jr Barth and R. J. Noelle, The role of CD40/CD154 interactions in the priming, differentiation, and effector function of helper and cytotoxic T cells, J. Leukoc. Biol., 63 (1998), 418-428. [24] J. E. Marsden and M. McKracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. [25] D. Mucida, M. M. Husain, S. Muroi et. al, Transcriptional reprogramming of mature CD4$^+$ helper T cells generates distinct MHC class II-restricted cytotoxic T lymphocytes, Nat. Immunol., 14 (2013), 281-289. doi: 10.1038/ni.2523. [26] A. Perez-Diez, N. T. Joncker, K. Choi, W. F. N. Chan, C. C. Anderson, O. Lantz and P. Matzinger, CD4 cells can be more efficient at tumor rejection than CD8 cells, Blood, 109 (2007), 5346-5354. [27] S. A. Quezada, T. R. Simpson, K. S. Peggs, T. Merghoub, J. Vider, X. Z. Fan, R. Blasberg, H. Yagita, P. Muranski, P. A. Antony, N. P. Restifo and J. P. Allison, Tumor-reactive CD4(+) T cells develop cytotoxic activity and eradicate large established melanoma after transfer into lymphopenic hosts, J. Exp. Med., 207 (2010), 637-650. doi: 10.1084/jem.20091918. [28] S. A. Rosenberg, Progress in human tumour immunology and immunotherapy, Nature, 411 (2001), 380-384. [29] S. Ruan and G. S. K. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay, J. Math. Anal. Appl., 204 (1996), 786-812. doi: 10.1006/jmaa.1996.0468. [30] K. E. Starkov and L. N. Coria, Global dynamics of the Kirschner-Panetta model for the tumor immunotherapy, Nonlinear Anal. Real World Appl., 14 (2013), 1425-1433. doi: 10.1016/j.nonrwa.2012.10.006. [31] E. Stockert, E. Jager, Y. T. Chen, M. J. Scanlan, I. Gout, J. Karbach, M. Arand, A. Knuth and L. J. Old, A survey of the humoral immune response of cancer patients to a panel of human tumor antigens, J. Exp. Med., 187 (1998), 1349-1354. doi: 10.1084/jem.187.8.1349. [32] Z. Szymanska, Analysis of immunotheray models in the context of cancer dynamics, Int. J. Appl. Math. Comput. Sci., 13 (2003), 407-418. [33] A. Tsygvintsev, S. Marino and D. E. Kirschner, A mathematical model of gene therapy for the treatment of cancer, in Mathematical Methods and Models in Biomedicine (eds. U. Ledzewicz, H. Schattler, A. Friedman and E. Kashdan), Springer New York, 2013, 367-385. doi: 10.1007/978-1-4614-4178-6_13. [34] M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, J. Math. Biol., 47 (2003), 270-294. doi: 10.1007/s00285-003-0211-0. [35] L. Zamai, C. Ponti, P. Mirandola, G. Gobbi, S. Papa, L. Galeotti, L. Cocco and M. Vitale, NK cells and cancer, J. Immunol., 178 (2007), 4011-4016.

show all references

##### References:
 [1] J. A. Adam and N. Bellomo, A Survey of Models for Tumor-immune System Dynamics, Birkhauser, Boston, 1997. doi: 10.1007/978-0-8176-8119-7. [2] J. C. Arciero, T. L. Jackson and D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 39-58. [3] T. Boon and P. Van der Bruggen, Human tumor antigens recognized by T lymphocytes, J. Exp. Med., 183 (1996), 725-729. doi: 10.1084/jem.183.3.725. [4] C. Bourgeois, B. Rocha and C. Tanchot, A role for CD40 expression on CD8$^+$ T cells in the generation of CD8$^+$ T cell memory, Science, 297 (2002), 2060-2063. doi: 10.1126/science.1072615. [5] L. G. de Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res., 65 (2005), 7950-7958. [6] L. G. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations, J. Theor. Biol., 238 (2006), 841-862. doi: 10.1016/j.jtbi.2005.06.037. [7] K. E. de Visser, A. Eichten and L. M. Coussens, Paradoxical roles of the immune system during cancer development, Nat. Rev. Cancer, 6 (2006), 24-37. [8] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences, Phys. D, 208 (2005), 220-235. doi: 10.1016/j.physd.2005.06.032. [9] R. Eftimie, J. L. Bramson and D. J. D. Earn, Modeling anti-tumor Th1 and Th2 immunity in the rejection of melanoma, J. Theor. Biol., 265 (2010), 467-480. doi: 10.1016/j.jtbi.2010.04.030. [10] R. Eftimie, J. L. Bramson and D. J. D. Earn, Interactions between the immune system and cancer: A brief review of non-spatial mathematical models, Bull. Math. Biol., 73 (2011), 2-32. doi: 10.1007/s11538-010-9526-3. [11] B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718195. [12] P. Fortin and M. C. Mackey, Periodic chronic myelogenous leukaemia: spectral analysis of blood cell counts and aetiological implications, Brit. J. Haematol., 104 (1999), 336-345. doi: 10.1046/j.1365-2141.1999.01168.x. [13] M. Galach, Dynamics of the tumor-immune system competition-the effect of time delay, Int. J. Appl. Math. Comput. Sci., 13 (2003), 395-406. [14] K. Hung, R. Hayashi, A. Lafond-Walker, C. Lowenstein, D. Pardoll and H. Levitsky, The central role of CD4+ T cells in the antitumor immune response, J. Exp. Med., 188 (1998), 2357-2368. doi: 10.1084/jem.188.12.2357. [15] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252. doi: 10.1007/s002850050127. [16] D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Math. Biosci. Eng., 6 (2009), 573-583. doi: 10.3934/mbe.2009.6.573. [17] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 2 (1994), 295-321. [18] Y. P. Lai, C. J. Jeng and S. C. Chen, The roles of CD4$^+$ T cells in tumor immunity, ISRN Immunology, 2011 (2011), 1-6. doi: 10.5402/2011/497397. [19] C. Letellier, F. Denis and L. A. Aguirre, What can be learned from a chaotic cancer model? J. Theor. Biol., 322 (2013), 7-16. doi: 10.1016/j.jtbi.2013.01.003. [20] O. Lejeune, M. A. J. Chaplain and I. EI Akili, Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumours, Math. Comput. Modelling, 47 (2008), 649-662. doi: 10.1016/j.mcm.2007.02.026. [21] D. Liu, S. Ruan and D. Zhu, Bifurcation analysis in models of tumor and immune system interactions, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 151-168. doi: 10.3934/dcdsb.2009.12.151. [22] D. Liu, S. Ruan and D. Zhu, Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions, Math. Biosci. Eng., 9 (2012), 347-368. doi: 10.3934/mbe.2012.9.347. [23] M. F. Mackey, R. J. Jr Barth and R. J. Noelle, The role of CD40/CD154 interactions in the priming, differentiation, and effector function of helper and cytotoxic T cells, J. Leukoc. Biol., 63 (1998), 418-428. [24] J. E. Marsden and M. McKracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. [25] D. Mucida, M. M. Husain, S. Muroi et. al, Transcriptional reprogramming of mature CD4$^+$ helper T cells generates distinct MHC class II-restricted cytotoxic T lymphocytes, Nat. Immunol., 14 (2013), 281-289. doi: 10.1038/ni.2523. [26] A. Perez-Diez, N. T. Joncker, K. Choi, W. F. N. Chan, C. C. Anderson, O. Lantz and P. Matzinger, CD4 cells can be more efficient at tumor rejection than CD8 cells, Blood, 109 (2007), 5346-5354. [27] S. A. Quezada, T. R. Simpson, K. S. Peggs, T. Merghoub, J. Vider, X. Z. Fan, R. Blasberg, H. Yagita, P. Muranski, P. A. Antony, N. P. Restifo and J. P. Allison, Tumor-reactive CD4(+) T cells develop cytotoxic activity and eradicate large established melanoma after transfer into lymphopenic hosts, J. Exp. Med., 207 (2010), 637-650. doi: 10.1084/jem.20091918. [28] S. A. Rosenberg, Progress in human tumour immunology and immunotherapy, Nature, 411 (2001), 380-384. [29] S. Ruan and G. S. K. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay, J. Math. Anal. Appl., 204 (1996), 786-812. doi: 10.1006/jmaa.1996.0468. [30] K. E. Starkov and L. N. Coria, Global dynamics of the Kirschner-Panetta model for the tumor immunotherapy, Nonlinear Anal. Real World Appl., 14 (2013), 1425-1433. doi: 10.1016/j.nonrwa.2012.10.006. [31] E. Stockert, E. Jager, Y. T. Chen, M. J. Scanlan, I. Gout, J. Karbach, M. Arand, A. Knuth and L. J. Old, A survey of the humoral immune response of cancer patients to a panel of human tumor antigens, J. Exp. Med., 187 (1998), 1349-1354. doi: 10.1084/jem.187.8.1349. [32] Z. Szymanska, Analysis of immunotheray models in the context of cancer dynamics, Int. J. Appl. Math. Comput. Sci., 13 (2003), 407-418. [33] A. Tsygvintsev, S. Marino and D. E. Kirschner, A mathematical model of gene therapy for the treatment of cancer, in Mathematical Methods and Models in Biomedicine (eds. U. Ledzewicz, H. Schattler, A. Friedman and E. Kashdan), Springer New York, 2013, 367-385. doi: 10.1007/978-1-4614-4178-6_13. [34] M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, J. Math. Biol., 47 (2003), 270-294. doi: 10.1007/s00285-003-0211-0. [35] L. Zamai, C. Ponti, P. Mirandola, G. Gobbi, S. Papa, L. Galeotti, L. Cocco and M. Vitale, NK cells and cancer, J. Immunol., 178 (2007), 4011-4016.
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2020 Impact Factor: 1.327


text:
# Finding the radius of a circle in terms of other radii.

Question: Three circles with centers respectively $A,B,C$ are mutualy tangent. Express the radius of circle with center A in terms of BC, AC, and AB respectively.

picture of tangent circles

My work: Let circles A, B, and C have radii a, b, and c, respectively and let $AB=x, AC=y, BC=z$. From here, I am not sure what to do. I tried systems of equations, but it didn't work.

Could someone please tell me what to do from here so that I can get the radius of circle A (a) and the reason behind it. I understand that there is a guess and check method to get the correct answer. However, it would be more helpful if someone could explain the rest of the algebra that I have to do to get to the answer.

Hint

$$\begin{cases}a+b=|AB|\\ a+c=|AC| \\ b+c=|BC|\end{cases}$$

$AC=r_a+r_c$

$AB=r_a+r_b$

$BC=r_b+r_c$

Subtract third equation from the first two.

$2r_a=AC+AB-BC$

$r_a=\frac{AC+AB-BC}{2}$

as you have assumed that a,b,c are the radius of the circles A, B, C respectively

you can say that $$AB=a+b$$

$$AC=a+c$$

$$BC=b+c$$

thus $BC-b=c$

put in the second equation

$AC=a+(BC-b)$

$AC-BC-a=-b$

$BC-AC+a=b$

put in equation 1

you get $AB=a+(BC-AC+a)$

$$2a=AB-BC+AC$$

$$a=\frac{AB-BC+AC}{2}$$

as equation 1 says $AB=a+b$

thus put a and get the value of b

similarly, solve for b and c

• Hi James, thanks for helping me. From here, what should I do to find a, which is the radius of circle A? – T.Mok Aug 15 '18 at 14:19
• see the edited answer three variables and three equations – Deepesh Meena Aug 15 '18 at 14:24


text:
# Boundary Conditions#

Run this notebook in your browser using Binder.

This notebook will give a tutorial on setting up boundary conditions in Tidy3d.

[1]:

# standard python imports
import numpy as np
import matplotlib.pyplot as plt

# tidy3d imports
import tidy3d as td
import tidy3d.web as web

## Define Simulation Parameters#

First, we’ll define some basc simulation parameters, the size of the domain, and the discretization resolution.

[2]:

# Define material properties
medium = td.Medium(permittivity=2)

wavelength = 1
f0 = td.C_0 / wavelength / np.sqrt(medium.permittivity)

# Set the domain size in x, y, and z
domain_size = 12 * wavelength

# create the geometry
geometry = []

# construct simulation size array
sim_size = (domain_size, domain_size, domain_size)

# Bandwidth in Hz
fwidth = f0 / 40.0

# Gaussian source offset; the source peak is at time t = offset/fwidth
offset = 4.

# time dependence of sources
source_time = td.GaussianPulse(freq0=f0, fwidth=fwidth, offset=offset)

# Simulation run time past the source decay (around t=2*offset/fwidth)
run_time = 40 / fwidth

## Create sources and monitors#

To study the effect of the various boundary conditions, we’ll define a point dipole source and a series of frequency- and time-domain monitors in the volume of the simulation domain and at its edges.

[3]:

# create a point dipole source
dipole = td.PointDipole(
center=(0,0,0),
source_time=source_time,
polarization="Ex",
)

# these monitors will be used to plot fields on planes through the middle of the domain in the frequency domain
monitor_xz_freq = td.FieldMonitor(
center=(0, 0, 0),
size=(domain_size, 0, domain_size),
freqs=[f0],
name='xz_freq')
monitor_yz_freq = td.FieldMonitor(
center=(0, 0, 0),
size=(0, domain_size, domain_size),
freqs=[f0],
name='yz_freq')
monitor_xy_freq = td.FieldMonitor(
center=(0, 0, 0),
size=(domain_size, domain_size, 0),
freqs=[f0],
name='xy_freq')

# this monitor will be used to plot fields on a plane through the middle of the domain in the time domain
monitor_xz_time = td.FieldTimeMonitor(
center=(0, 0, 0),
size=(domain_size, 0, domain_size),
interval=50,
name='xz_time')

## Boundary specifications#

A BoundarySpec <https://docs.simulation.cloud/projects/tidy3d/en/latest/_autosummary/tidy3d.BoundarySpec.html>__ object defines the boundary conditions applied on each of the 6 domain edges, and is provided as an input to the simulation. In the following sections, we’ll explore several different features within BoundarySpec <https://docs.simulation.cloud/projects/tidy3d/en/latest/_autosummary/tidy3d.BoundarySpec.html>__ and different ways of defining it.

A BoundarySpec <https://docs.simulation.cloud/projects/tidy3d/en/latest/_autosummary/tidy3d.BoundarySpec.html>__ consists of three Boundary <https://docs.simulation.cloud/projects/tidy3d/en/latest/_autosummary/tidy3d.Boundary.html>__ objects, each defining the boundaries on the plus and minus side of each dimension. A number of convenience functions are available to quickly define various types of boundaries, which will be demonstrated below.

### Example 1: Default PML boundaries along some dimensions#

In most cases, one just wants to specify whether there are absorbing PML layers along any of the x, y, z dimensions. To do this, we can call the BoundarySpec.pml(x=False, y=False, z=False) method to construct boundary conditions, supplying True along a dimension where we want PML.

Let’s try putting pml on only the x edge.

[4]:

# define a basic boundary spec setting PML in along x only
bspec_pml = td.BoundarySpec.pml(x=True)

sim = td.Simulation(
size=sim_size,
sources=[dipole],
monitors=[monitor_xz_time],
run_time=run_time,
boundary_spec=bspec_pml,
)

ax = sim.plot(y=0)

[09:51:30] INFO     Auto meshing using wavelength 1.4142 defined from sources.                     grid_spec.py:473

Now, let’s test by placing PMLs along all edges using the convenience method all_sides() and running the simulation.

[5]:

# define a basic boundary spec setting PML in all directions
bspec_pml = td.BoundarySpec.all_sides(boundary=td.PML())

# initialize the simulation object with the above boundary spec and source
sim = td.Simulation(
size=sim_size,
sources=[dipole],
monitors=[monitor_xz_time],
run_time=run_time,
boundary_spec=bspec_pml,
)

# Visualize the geometry
fig, ax1 = plt.subplots(figsize=(4, 4))
sim.plot(y=0, ax=ax1);

           INFO     Auto meshing using wavelength 1.4142 defined from sources.                     grid_spec.py:473

[6]:

sim_data = web.run(sim, task_name='bc_example1', path='data/bc_example1.hdf5')

↓ monitor_data.hdf5 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╸━━ 94.4% • 8.9/9.4 MB • 4.2 MB/s • 0:00:01

[09:52:12] INFO     loading SimulationData from data/bc_example1.hdf5                                 webapi.py:397

#### Visualize results#

We can observe the effect of the PML by looking at the fields in the time domain as they impinge on the boundaries. The figure shows that the fields are absorbed by the PML layers as expected, and no reflections are observed.

[7]:

fig, ax = plt.subplots(tight_layout=True, figsize=(5, 4))
sim_data.plot_field(field_monitor_name='xz_time', field_name='Ex', y=0, val='abs', t=2e-13, ax=ax)

plt.show()

           INFO     Auto meshing using wavelength 1.4142 defined from sources.                     grid_spec.py:473

### Example 2: different boundaries on different edges#

Here, we’ll place a PML along the y and z directions. In the x direction, we’ll place a PML on the left side (x-minus), and a PEC on the right (x-plus). To specify individual boundary conditions along different dimensions, instead of BoundarySpec <https://docs.simulation.cloud/projects/tidy3d/en/latest/_autosummary/tidy3d.BoundarySpec.html>__, the class Boundary <https://docs.simulation.cloud/projects/tidy3d/en/latest/_autosummary/tidy3d.Boundary.html>__ is used, which defines the plus and minus boundaries along a single dimension.

We’ll test this set of boundaries by placing an angled Gaussian beam near the lower edge of the domain and observing the field bounce off the PEC in x-plus.

[8]:

# this defines a 1D boundary with a PML on both the plus and minus sides, which can be reused for both y and z directions
boundary_yz = td.Boundary.pml(num_layers=15)

# this defines a 1D boundary with a PML on the minus side and a PEC on the plus side
boundary_x = td.Boundary(minus=td.PML(), plus=td.PECBoundary())

# now just set these in the boundary spec along the appropriate dimensions
bspec_pml_pec = td.BoundarySpec(
x=boundary_x,
y=boundary_yz,
z=boundary_yz
)

# create the Gaussian beam source
buffer_source = domain_size / 10 # distance between the source and the bottom of the domain
gaussian_beam = td.GaussianBeam(
center=(0,0,-domain_size/2 + buffer_source),
size=(td.inf, td.inf, 0),
source_time=source_time,
direction='+',
pol_angle=0,
angle_theta=np.pi / 4.0,
angle_phi=np.pi / 8.0,
waist_distance=-wavelength * 4
)

# initialize the simulation object with the above boundary spec and source
sim = td.Simulation(
size=sim_size,
sources=[gaussian_beam],
monitors=[monitor_xz_time],
run_time=run_time,
boundary_spec=bspec_pml_pec
)

We can verify that the PML is placed only on the left hand side and not on the right.

[9]:

# Visualize the geometry
fig, ax1 = plt.subplots(figsize=(4, 4))
sim.plot(y=0, ax=ax1);

           INFO     Auto meshing using wavelength 1.4142 defined from sources.                     grid_spec.py:473

#### Run Simulation#

[10]:

sim_data = web.run(sim, task_name='bc_example2', path='data/bc_example2.hdf5')

↓ monitor_data.hdf5 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╸━ 97.2% • 9.0/9.3 MB • 828.3 kB/s • 0:00:01

[09:53:56] INFO     loading SimulationData from data/bc_example2.hdf5                                 webapi.py:397

#### Visualize results#

We can observe the effect of the PEC on x-plus by looking at the fields in the time domain as they bounce off the PEC boundary, as shown in the figure. Furthermore, the z-component of the electric field, which is tangential to the PEC boundary, goes to 0 at the boundary as expected.

[11]:

fig, (ax1, ax2) = plt.subplots(1, 2, tight_layout=True, figsize=(10, 4))
sim_data.plot_field(field_monitor_name='xz_time', field_name='Ex', y=0, val='abs', t=2e-13, ax=ax1)
sim_data.plot_field(field_monitor_name='xz_time', field_name='Ez', y=0, val='abs', t=2e-13, ax=ax2)
plt.show()

           INFO     Auto meshing using wavelength 1.4142 defined from sources.                     grid_spec.py:473

### Example 3: different boundaries on different edges#

Next, let’s consider an even more general setup where all 6 boundaries of the domain are individually specified, and different types of boundaries are mixed in the plus and minus sides along each dimension.

We’ll test this set of boundaries by placing the point dipole at the center of the domain again, this time studying the fields at the edges of the domain and checking that the various tangential field components satisfy the PEC and PMC boundary conditions as expected from theory.

[12]:

# define the boundary spec
bspec_general = td.BoundarySpec(
x=td.Boundary(minus=td.Periodic(), plus=td.PECBoundary()),
y=td.Boundary(minus=td.PECBoundary(), plus=td.PMCBoundary()),
z=td.Boundary(minus=td.Absorber(), plus=td.PML())
)
# note that when a periodic boundary is applied across from a PEC (PMC), the periodic boundary is just replaced by the PEC (PMC).

# initialize the simulation object with the above boundary spec and the dipole source from earlier
sim = td.Simulation(
size=sim_size,
sources=[dipole],
monitors=[monitor_xz_freq, monitor_yz_freq, monitor_xy_freq],
run_time=run_time,
boundary_spec=bspec_general,
)

# Visualize the geometry
fig, ax1 = plt.subplots(figsize=(4, 4))
sim.plot(x=0, ax=ax1);

[09:53:57] WARNING  A periodic boundary condition was specified on the opposide side of a perfect   boundary.py:401
electric or magnetic conductor boundary. This periodic boundary condition will
be replaced by the perfect electric or magnetic conductor across from it.

           INFO     Auto meshing using wavelength 1.4142 defined from sources.                     grid_spec.py:473

#### Run Simulation#

[13]:

sim_data = web.run(sim, task_name='bc_example3', path='data/bc_example3.hdf5')

↓ monitor_data.hdf5 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╸━━━━━━━━━ 77.0% • 1.8/2.4 MB • 2.7 MB/s • 0:00:01

[09:54:42] INFO     loading SimulationData from data/bc_example3.hdf5                                 webapi.py:397

#### Visualize results#

In x-plus, we have a PEC, so Ey and Ez should be nearly zero on the x-plus boundary.

In x-minus we have a periodic condition across from the PEC in x-plus, which just means that there’s also a PEC in x-minus. So again Ey and Ez should be nearly zero on the x-minus boundary.

In y-minus, we have another PEC, so Ex and Ez should be nearly zero on the y-minus boundary.

In y-plus, we have a PMC, so Hx and Hz should be nearly zero on the y-plus boundary.

Each of these cases is verified below.

[14]:

fig, (ax1, ax2) = plt.subplots(1, 2, tight_layout=True, figsize=(10, 4))
sim_data.plot_field(field_monitor_name='xz_freq', field_name='Ey', y=0, val='abs', freq=f0, ax=ax1)
sim_data.plot_field(field_monitor_name='xz_freq', field_name='Ez', y=0, val='abs', freq=f0, ax=ax2)
fig.suptitle('Notice that both Ey and Ez go to zero on both sides', fontsize=16)

fig, (ax1, ax2) = plt.subplots(1, 2, tight_layout=True, figsize=(10, 4))
sim_data.plot_field(field_monitor_name='yz_freq', field_name='Ex', x=0, val='abs', freq=f0, ax=ax1)
sim_data.plot_field(field_monitor_name='yz_freq', field_name='Ez', x=0, val='abs', freq=f0, ax=ax2)
fig.suptitle('Notice that both Ex and Ez go to zero on the left (PEC) side', fontsize=16)

fig, (ax1, ax2) = plt.subplots(1, 2, tight_layout=True, figsize=(10, 4))
sim_data.plot_field(field_monitor_name='yz_freq', field_name='Hx', x=0, val='abs', freq=f0, ax=ax1)
sim_data.plot_field(field_monitor_name='yz_freq', field_name='Hz', x=0, val='abs', freq=f0, ax=ax2)
fig.suptitle('Notice that both Hx and Hz go to zero on the right (PMC) side', fontsize=16)
plt.show()

           INFO     Auto meshing using wavelength 1.4142 defined from sources.                     grid_spec.py:473

           WARNING  'freq' suppled to 'plot_field', frequency selection key renamed to 'f' and      sim_data.py:316
'freq' will error in future release, please update your local script to use
'f=value'.

[09:54:43] WARNING  'freq' suppled to 'plot_field', frequency selection key renamed to 'f' and      sim_data.py:316
'freq' will error in future release, please update your local script to use
'f=value'.

           WARNING  'freq' suppled to 'plot_field', frequency selection key renamed to 'f' and      sim_data.py:316
'freq' will error in future release, please update your local script to use
'f=value'.

           WARNING  'freq' suppled to 'plot_field', frequency selection key renamed to 'f' and      sim_data.py:316
'freq' will error in future release, please update your local script to use
'f=value'.

           WARNING  'freq' suppled to 'plot_field', frequency selection key renamed to 'f' and      sim_data.py:316
'freq' will error in future release, please update your local script to use
'f=value'.

           WARNING  'freq' suppled to 'plot_field', frequency selection key renamed to 'f' and      sim_data.py:316
'freq' will error in future release, please update your local script to use
'f=value'.

### Example 4: Bloch boundary conditions along x and y to allow injecting an angled plane wave source#

Finally, we’ll use the Bloch boundary condition in combination with a plane wave source to demonstrate the injection of an angled plane wave. The Bloch boundaries are used along the x and y dimensions, while PMLs are used along z. The Bloch vectors are automatically computed based on the angles defined in the plane wave source. A dielectric background medium with a relative permittivity of 2 is used.

We’ll test this set of boundaries by placing an angled plane wave near the lower edge of the domain and observing the plane wave in the frequency domain.

[15]:

# First, define the plane wave source, since it is needed to define the Bloch boundary in this case.
# Note that in general, the Bloch boundary can also be defined by just providing a bandstructure-normalized Bloch vector.

buffer_source = domain_size / 10 # distance between the source and the bottom of the domain
plane_wave = td.PlaneWave(
center=(0,0,-domain_size/2 + buffer_source),
size=(td.inf, td.inf, 0),
source_time=source_time,
direction='+',
pol_angle=0,
angle_theta=np.pi / 3.0,
angle_phi=np.pi / 6.0
)

# create the Bloch boundaries
bloch_x = td.Boundary.bloch_from_source(source=plane_wave, domain_size=sim_size[0], axis=0, medium=medium)
bloch_y = td.Boundary.bloch_from_source(source=plane_wave, domain_size=sim_size[1], axis=1, medium=medium)

bspec_bloch = td.BoundarySpec(
x=bloch_x,
y=bloch_y,
z=td.Boundary.pml()
)

# initialize the simulation object with the above boundary spec and source
sim = td.Simulation(
size=sim_size,
sources=[plane_wave],
monitors=[monitor_xz_freq, monitor_yz_freq, monitor_xy_freq],
run_time=run_time,
boundary_spec=bspec_bloch,
medium=medium
)

# Visualize the geometry
fig, ax1 = plt.subplots(figsize=(4, 4))
sim.plot(y=0, ax=ax1);

[09:54:44] INFO     Auto meshing using wavelength 1.4142 defined from sources.                     grid_spec.py:473

#### Run Simulation#

[16]:

sim_data = web.run(sim, task_name='bc_example4', path='data/bc_example4.hdf5')

↓ monitor_data.hdf5 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╸━━━━━ 87.5% • 3.9/4.5 MB • 3.7 MB/s • 0:00:01

[09:55:27] INFO     loading SimulationData from data/bc_example4.hdf5                                 webapi.py:397

#### Visualize results#

In shown in the plots, the plane wave is successfully injected at the specified angles without reflections.

[17]:

fig, (ax1, ax2, ax3) = plt.subplots(1, 3, tight_layout=True, figsize=(13, 4))
sim_data.plot_field(field_monitor_name='xz_freq', field_name='Ey', y=0, val='real', f=f0, ax=ax1)
sim_data.plot_field(field_monitor_name='xz_freq', field_name='Ey', y=0, val='real', f=f0, ax=ax2)
sim_data.plot_field(field_monitor_name='xz_freq', field_name='Ez', y=0, val='real', f=f0, ax=ax3)

fig, (ax1, ax2, ax3) = plt.subplots(1, 3, tight_layout=True, figsize=(13, 4))
sim_data.plot_field(field_monitor_name='yz_freq', field_name='Ex', x=0, val='real', f=f0, ax=ax1)
sim_data.plot_field(field_monitor_name='yz_freq', field_name='Ey', x=0, val='real', f=f0, ax=ax2)
sim_data.plot_field(field_monitor_name='yz_freq', field_name='Ez', x=0, val='real', f=f0, ax=ax3)

fig, (ax1, ax2, ax3) = plt.subplots(1, 3, tight_layout=True, figsize=(13, 4))
sim_data.plot_field(field_monitor_name='xy_freq', field_name='Ex', z=0, val='real', f=f0, ax=ax1)
sim_data.plot_field(field_monitor_name='xy_freq', field_name='Ey', z=0, val='real', f=f0, ax=ax2)
sim_data.plot_field(field_monitor_name='xy_freq', field_name='Ez', z=0, val='real', f=f0, ax=ax3)
fig.suptitle('Observe the phi=pi/6 angle w.r.t. the x axis', fontsize=16)

plt.show()

[09:55:28] INFO     Auto meshing using wavelength 1.4142 defined from sources.                     grid_spec.py:473

## Summary#

• The fastest way to define boundary conditions with periodic and absorbing boundaries on some sides is through td.BoundarySpec.pml(y=True) (i.e., for PML along y only), or through td.BoundarySpec.all_sides(boundary=PML()) to set the same boundary on all sides (in this example, PML on all sides).

• To further fine tune the global boundary specifications, use BoundarySpec(x, y, z), where x, y, z are Boundary objects.

• Boundary objects have some convenience methods, like .pml() for defining the boundaries along an edge with more parameters.

• Boundary objects can be further specified by supplying boundary conditions to the plus and minus fields.

[ ]:


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# Tag Info

61

Are random errors necessarily gaussian? Errors are very often Gaussian, but not always. Here are some physical systems where random fluctuations (or "errors" if you're in a context with the thing that's varying constitutes an error) are not Gaussian: The distribution of times between clicks in a photodetector exposed to light is an exponential distribution....

57

First, distributions are not always bell-shaped. A very important set of distributions decrease from a maximum at $x=0$, such as the exponential distribution (delay times until a random event such as a radioactive decay) or power-laws (size distributions of randomly fragmenting objects, earthquakes, ore grade, and many other things). Stable distributions ...

48

The comment Samuel Weir makes on the fine structure constant is pretty close to an answer. For electromagnetic transitions of the nucleus, these would change if the fine structure constant changed over time. Yet spectral data on distant sources indicates no such change. The atomic transitions would change their energies and we would observe photons from ...

40

Can the Second Law of Thermodynamics / Entropy override Newton's Laws? No. In the example given, every particle obeys Newton's laws. There is no particle that is not obeying $F=ma$. From the example below, it seems that there is an underlying "Force" behind the second law of thermodynamics which drives it and which is more powerful than any other law and ...

33

There are various questions that one would have to answer, if one wished to claim that there had been large changes in decay rates over geological time. Here is what I think might be the best experiment to prove this claim. Without using radiological evidence, one can deduce that the Earth is at least a billion years old by counting annual sedimentation ...

32

The surprising answer is that nothing triggers it. In quantum mechanics all we can talk about is the probabilities of various events happening: whether they actually happen in a given period is truly random. There is no secret mechanism which we could find which controls whether an event happens or not. Well, there are, really, three-and-a-half ...

31

The basic point here is that we don't "know" anything about "the real world". All we have is a model of the world, and some measure of how well the model matches what we observe. Of course, you can construct an entirely consistent model which says "an invisible, unobservable entity created everything I have ever observed one second before I was born, and ...

17

‘Bell curve’ often refers to a Gaussian distribution. That distribution is so common that it’s also called the normal distribution. It’s very common because it emerges any time you’re looking at the sum of many things from a single distribution: I.e. lots of tiny fluctuations which, under the Central Limit Theorem, add up to a Gaussian distribution. ...

15

The reason is probably the central limit theorem: When you add lots of independent random variables, their sum will form a normal distribution, irrespective of their individual probability distributions. This makes normal distributions a pretty good guess if you do not have information about the origin of the error or if you have multiple sources of error. ...

15

Averaging destroys information. Do it as late as is practical in your analysis. A commenter points out that, if you are making many position/velocity measurements of the same object as it moves once, a simple linear regression is more robust as a velocity estimator than an ensemble of point-by-point analyses.

14

13

Yes, the only sensible formula for the total error is the sum in quadrature, $$\Delta X_{\rm total} = \sqrt { \Delta X_{\rm syst}^2 + \Delta X_{\rm stat}^2 }$$ The key assumption behind the validity of the formula is that the two sources of error are independent i.e. uncorrelated. $$\langle \Delta X_{\rm syst} \Delta X_{\rm stat} \rangle = 0$$ Because of ...

12

Geant is a framework---which means that you use it to build applications that simulate the detector and physics you are interested in. The simulation can include all of physics and the complete detector including electronics and trigger (i.e. you can write your simulation so that it output a data file that looks just like the one you are going to get from ...

12

Actually a paper recently came out, and highlighted in Popular Science, discussing using fermionic field concepts to model crowd avoidance at Netflix. You can imagine that the same concept could be used to consider in any situation where there are large numbers of people competing for limited preferred items. Update Now that we have a few minutes, rather ...

12

You might like this 110-page paper by me and Alex Arkhipov, which is all about a quantum bosonic analogue of Galton's board (we even use the same graphic you did -- see Section 1.1!). In particular, we gave strong evidence that such a board (with an arbitrary configuration of "pegs," and with multiple entry points for the "balls") is exponentially hard even ...

12

Here is how I interpret what happened: You used Excel to compute the coefficients of the Gaussian that best describe the data: mean $\mu$, standard deviation $\sigma$, and magnitude $A$ for a curve $$Y=Ae^{-(x-\mu)^2/2\sigma^2}$$ Then you evaluated that function at a number of X values. Since the X values are not symmetrical about the calculated mean, you ...

12

There is no restriction of QM avoiding this problem even if these states appear to be a bit weird since they have no "preferred" spatial localization, but in principle they cannot be excluded. I stress that we are discussing about proper states, i.e., elements of $L^2(\mathbb R)$ and not, for instance, eigenfunctions of the momentum operator. Actually, the ...

11

The idea is just that if the uncertainties are small enough you can approximate the function by its Taylor series $$f(x_i + \delta_i) \approx f(x_i) + \sum_j \frac{\partial f(x_i)}{\partial x_j} \delta_j + \sum_{j,k} \frac{1}{2} \frac{\partial^2 f(x_i)}{\partial x_j \partial x_k} \delta_j \delta_k + \cdots.$$ If you neglect the second order terms the ...

11

Answers here have generally addressed the different question of whether empirical variables should be Gaussian, but 21joanna12 asked about experimental errors, which admit a completely different analysis. The best resource on that question I can recommend is Chapter 7 of Probability Theory: The Logic of Science by E T Jaynes. In short, there are good reasons ...

11

The thermal motion has to be taken into account in any understanding of the example. It’s just as important as gravity. if there was no gravity no gravitational force, zero gravitational potential) the gases would fully mix if the temperature was zero (no thermal energy, no thermal motion) the gases would fully separate out. When both effects are present,...

10

The formula you've specified $$\Delta k = \sqrt{(\Delta k_1)^2 + (\Delta k_2)^2}$$ is the formula to obtain error of quantity $k$, as being dependent on $k_1$ and $k_2$ according to the following expression $$k = k_1 + k_2.$$ Generally, to obtain experimental error of a dependent quantity (and the expression stated in your question), you start with ...

10

On the deepest level, particles are indistinguishable if and only if they have the same quantum numbers (mass, spin, and charges). However, in statistical mechanics one often studies effective theories where there are additional means of distinguishing particles. Two important examples: In modeling molecular fluids, two atoms on the same molecule are ...

10

This is something that particle physicists are perfectly well aware of. For any given observed effect, there is always a nonzero probability that the observation will be a false positive that was caused by a random fluctuation. The name of the game is taking enough data that this probability is small enough. In general, the more data you take, the less ...

9

If there are enough data and the prior is not completely unreasonable, the frequentist and the Bayesian approach give essentially the same answer. This is related to the central limit theorem. If data are fairly scarce, the two approaches may differ a lot. In this case the Bayesian approach is far preferable but only if the prior reflects true prior ...

9

A chi-squared test is used to compare binned data (e.g. a histogram) with another set of binned data or the predictions of a model binned in the same way. A K-S test is applied to unbinned data to compare the cumulative frequency of two distributions or compare a cumulative frequency against a model prediction of a cumulative frequency. Both chi-squared ...

9

Half life is, by definition, the amount of time until half of an infinitely large sample would decay. That's precisely equivalent (according to the frequentist interpretation of probability, if that matters to you) to the time until an individual particle's probability of decay reaches one half. The half life is a theoretical quantity that doesn't depend on ...

9

Model two consecutive measurements as the real values plus some noise. Call the first measured temperature $T_1$ and the second $T_2$. Call the measured noises $\gamma_1$ and $\gamma_2$, and suppose that they are drawn from a distribution $\Gamma(\gamma)$ and are uncorrelated. The (approximation to the) derivative is \text{Derivative} \approx \frac{(T_2 +...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
you are required to design a wind turbine with the following specifications and using the given $$C_{p}-\lambda$$ curve.

(i) electrical output of the turbine is 1.5MW at a wind velocity of 12m/s

(ii) the turbine tracks peak power from 4m/s to 12m/s

(iii) beyond 12m/s, power is held constnat at 1.5MW through pitch control

Provide the following design parameters and preformance characteristics

(i) swept area and blade diameter

(ii) estimated hub height

(iii) estimated gear ratio such that the generator speed is about 2000 rpm at a wind velocity of 12m/s

(iv)plot the electrical power output of the turbine against rpm for different wind velocities (6m/s, 8m/s, 10m/s, 12m/s, 14m/s, 16m/s) and show the operating regions of the turbine in the plot

(v)Plot the operating regions of the turbine against wind velocity ( asume cut-in speed =6m/s, cut-out speed=16m/s)

You may assume:

1. air density= 1.223 kg/m^3

2. overall system efficiency (mechanical+ electrical) = 90%


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## pottersheep Group Title Please help! Show that each statement is true (LOGS) [1/log base 5 of a] + [1/log base 3 of a] = [1/logbase 15a] one year ago one year ago

1. pottersheep

sorry last one is log base 15 of a

2. satellite73

3. satellite73

$\frac{\log_3(a)+\log_5(a)}{\log_3(a)\times \log_5(a)}$

4. satellite73

now lets flip it and see if we can show it is the same as $$\log_{15}(a)$$

5. satellite73

that is, can we show $\frac{\log_3(a)\times \log_5(a)}{\log_3(a)+\log_5(a)}=\log_15(a)$ and i think we can do it using the change of base formula and writing everything in terms of $$\log_{15}(x)$$

6. satellite73

do you know the change of base formula?

7. satellite73

you get a nasty compound fraction on the left, i will see if i can write it $\frac{\frac{\log_{15}(a)}{\log_{15}(5)}\times \frac{\log_{15}(a)}{\log_{15}(5)}}{\frac{\log_{15}(a)}{\log_{15}(5)}+\frac{\log_{15}(a)}{\log_{15}(3)}}$

8. satellite73

to clear the compound fraction, multiply top and bottom by $$\log_{15}(3)\log_{15}(5)$$

9. satellite73

the numerator will be $\log_{15}(a)\times \log_{15}(a)$ and the denominator will be $\log_{15}(a)\log_{15}(5)+\log_{15}(a)+\log_{15}(5)$ factor as $\log_{15}(a)\left(\log_{15}(5)+\log_{15}(3)\right)$ which gives $\log_{15}(a)\times \log_{15}(3\times 5)=\log_{15}(a)\times 1$

10. satellite73

typo above, second line should be $\log_{15}(a)\log_{15}(5)+\log_{15}(a)\log_{15}(5)$

11. satellite73

cancel and you get what you want

12. pottersheep

oooooo Thanks!


text:
# Online sexagesimal calculator tolek banan i inni online dating

Although today many use calculators and computers instead of abaci to calculate, abaci still remain in common use in some countries.

Merchants, traders and clerks in some parts of Eastern Europe, Russia, China and Africa use abaci, and they are still used to teach arithmetic to children.

The GPS coordinates are presented in the infowindow in an easy to copy and paste format.

You can also start to convert latitude and longitude by clicking on the map, which will pre-fill the fields with the GPS coordinates of the location you clicked on.

In any case, the address will not be geocoded automatically.

) \times b c_0 $$Example: Consider the number $$123_$$ (in base $$7$$), and apply the conversion algorithm:$$ 123 = \ \ N = 1 \ N = 1*7 2 = 9 \ N = 9*7 3 = 66 \ N = 123_ = 66_ So $$123_$$ is equal to $$66_$$ in base $$10$$.

The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of wood, stone, or metal. Some designs, like the bead frame consisting of beads divided into tens, are used mainly to teach arithmetic, although they remain popular in the post-Soviet states as a tool.

Other designs, such as the Japanese soroban, have been used for practical calculations even involving several digits.

In short: Decimal degrees = whole number of degrees, plus minutes divided by 60, plus seconds divided by 3600.

Now try to use our converter at the top of this page and you'll find that it's very easy accomplish this task.

### Search for online sexagesimal calculator:

(the exact shape of the Latin perhaps reflects the genitive form of the Greek word, ἄβακoς abakos).

## One thought on “online sexagesimal calculator”

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location:  Publications → journals
Search results

Search: MSC category 11J82 ( Measures of irrationality and of transcendence )

 Expand all        Collapse all Results 1 - 2 of 2

1. CMB 2005 (vol 48 pp. 147)

 Baker-Type Estimates for Linear Forms in the Values of $q$-Series We obtain lower estimates for the absolute values of linear forms of the values of generalized Heine series at non-zero points of an imaginary quadratic field~$\II$, in particular of the values of $q$-exponential function. These estimates depend on the individual coefficients, not only on the maximum of their absolute values. The proof uses a variant of classical Siegel's method applied to a system of functional Poincar\'e-type equations and the connection between the solutions of these functional equations and the generalized Heine series. Keywords:measure of linear independence, $q$-seriesCategories:11J82, 33D15
 Approximation algÃ©brique simultanÃ©e de nombres de Liouville The purpose of this paper is to show the limitations of the conjectures of algebraic approximation. For this, we construct points of $\bC^m$ which do not admit good algebraic approximations of bounded degree and height, when the bounds on the degree and the height are taken from specific sequences. The coordinates of these points are Liouville numbers. Category:11J82


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# non-P-points a Baire space

Let $X$ be a compact Hausdorff space. A P-point in $X$ is a point which does not lie in the boundary of the cozero set of a continuous real-valued function on $X$.

Question. Suppose that $X$ has no isolated points. Is the set of P-points a meagre subset of $X$? Failing this, is the set of non-P-points at least a Baire space?

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The set of non-P-points is a Baire space:

I´ll work with the usual definition of P-point: $p \in X$ is a P-point if the intersection of countably many neighborhoods of $p$ is again a neighborhood of $p$. This is equivalent to the definition in the question for compact $X$.

Fix a compact $X$, let $P \subseteq X$ be the set of all P-points and $N=X \setminus P$. The first thing to notice is that if $A$ is a $G_\delta$ subset of $X$ and $A \subseteq P$ then every $p \in A$ is isolated. This is because (using that $p$ is a P-point and regularity of $X$) there would be a closed (hence compact) neighborhood of $p$ consisting only of P-points. Since compact P-spaces are finite, we get that $p$ is isolated. It follows that if $X$ has no isolated points then $P$ contains no $G_\delta$ and in particular $N$ is dense in $X$.

Now if $\left< U_n : n\in \omega \right>$ is a sequence of open dense subsets of $N$, then each $U_n=V_n \cap N$ for some $V_n$ open in $X$. Since $N$ is dense in $X$ we have that $V_n$ must also be dense in $X$. Then $\bigcap_{n \in \omega}V_n$ is dense in $X$ and since $P$ contains no $G_\delta$, we get that $\bigcap_{n \in \omega}U_n=N \cap \bigcap_{n \in \omega}V_n$ is dense in $N$.

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Great! Many thanks. –  Douglas Somerset Apr 26 '12 at 17:10

The answer to the first half of your question is no, though, at least consistently.

Let $\mathbb{N}^*=\beta \mathbb{N} \setminus \mathbb{N}$ be the remainder of the Stone-Cech compactification of the integers. Under the Continuum Hypothesis, the set of all $P$-points is a dense subset of $\mathbb{N}^*$. But $\mathbb{N}^*$ has no dense meager subset. Indeed, let $\{N_n: n \in \mathbb{N} \}$ be a sequence of nowhere dense sets. Construct a decreasing sequence of open sets $\{U_n: n \in \mathbb{N} \}$ such that $U_n \cap D_n =\emptyset$ and $\overline{U_{n+1}} \subset U_n$ for every $n \in \mathbb{N}$. The set $\bigcap_{n \in \mathbb{N}} \overline{U_n}=\bigcap_{n \in \mathbb{N}} U_n$ is non-empty because of compactness. Moreover, every countable intersection of open subsets of $\mathbb{N}^*$ has non-empty interior (see, for example, Jan van Mill's chapter on $\beta \mathbb{N}$ in the Handbook of Set-theoretic Topology). Thus $Int(\bigcap_{n \in \mathbb{N}} U_n)$ is a non-empty open set missing every $N_n$ and hence $\bigcup_{n\in\mathbb{N}} N_n$ is not dense.

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Home > Circular Error > Circular Error Average

# Circular Error Average

## Contents

See the CEP literature overview for references and the shotGroups package for a free open source implementation: The general correlated normal estimator (DiDonato & Jarnagin, 1961a; Evans, 1985) is based on BURST RADIUS. Navigation errors generally follow a known error distribution and the uncertainty in position can be expressed as the probability that the error will not exceed a certain amount. To incorporate accuracy into the CEP concept in these conditions, CEP can be defined as the square root of the mean square error (MSE). Check This Out

It is based on the Pearson three-moment central $$\chi^{2}$$-approximation (Imhof, 1961; Pearson, 1959) of the cumulative distribution function of radial error in bivariate normal variables. The Grubbs-Pearson estimator has the theoretical advantage over the Grubbs-Patnaik estimator that the approximating distribution matches the true distribution not only in mean and variance but also in skewness. The smaller the PER, the more accurate the weapon. 50% of a weapons's "overs" (shells that fall beyond the target) and 50% of its "shorts" (shells that fall short of the In literature and in system/product specifications it can be found measurements of accuracy such as CEP, rms, Percentile 67%, Percentile 95%, 1 sigma, 2 sigma.

## Circular Error Probable

The Rayleigh estimator uses the Rayleigh quantile function for radial error (Culpepper, 1978; Singh, 1992). The general case allows that the point-of-aim is offset from the true center point-of-impact. Link to this page: circular-error average Facebook Twitter Feedback My bookmarks ? Change back to FALSE (to turn off Jumpstart) and it will revert back to normal calculations. ** Modeling Tip - If/Else is a powerful forumla.

H. (1966). "Asymptotic properties of some estimators of quantiles of circular error." Journal of the American Statistical Association, vol. 61 (315), pp. 618–632. That is, if CEP is n meters, 50% of rounds land within n meters of the target, 43% between n and 2n, and 7% between 2n and 3n meters, and the Percentiles can be determined by recognizing that the squared distance defined by two uncorrelated orthogonal Gaussian random variables (one for each axis) is chi-square distributed.[4] Approximate formulae are available to convert Spherical Error Probable Less used that the previous measurements are the: Mean Error: Average error.

The probability density function, the cumulative distribution function, and the quantile function are defined in closed form. Privacy policy About ShotStat Disclaimers Weapon Effectiveness Definitions5 March 2014 ReferencesArmies of NATO’s Central Front by David C. URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6217081&isnumber=6215928 External links Circular Error Probable in the Ballistipedia Retrieved from "https://en.wikipedia.org/w/index.php?title=Circular_error_probable&oldid=748558282" Categories: Applied probabilityMilitary terminologyAerial bombsArtillery operationBallisticsWeapon guidanceTheory of probability distributionsStatistical distance Navigation menu Personal tools Not logged inTalkContributionsCreate http://www.navipedia.net/index.php/Accuracy Your cache administrator is webmaster.

There is a correspondence between sigmas and percentiles. Circular Error Excel Your cache administrator is webmaster. Neither, it should be something in the middle because your cash blance grew during the month on its way to $200. If systematic accuracy bias is taken into account, this estimator becomes the Rice estimator. ## Circular Error Probable Calculation Other old, and less relevant approximations to the 50% quantile of the Hoyt distribution include Bell (1973), Nicholson (1974) and Siouris (1993). http://ballistipedia.com/index.php?title=Circular_Error_Probable Keyboard Word / Article Starts with Ends with Text A A A A Language: EnglishEspañolDeutschFrançaisItalianoالعربية中文简体PolskiPortuguêsNederlandsNorskΕλληνικήРусскийTürkçeאנגלית Twitter Get our app Log in / Register E-mail Password Wrong username or password. Circular Error Probable PER (PROBABLE ERROR RANGE). Circular Error Probable Excel It works best for a mostly circular distribution of $$(x,y)$$-coordinates (aspect ratio of data ellipse $$\leq 3$$). Further complications arise because some navigation systems are linear (one-dimensional) while others provide two or three dimensions of position[1]. his comment is here It is defined as the radius of a circle, centered on the mean, whose boundary is expected to include the landing points of 50% of the rounds.[2][3] That is, if a One shortcoming of the Grubbs estimators is that it is not possible to incorporate the confidence intervals of the variance estimates into the CEP estimate. This simulates as if an error was made and then corrected. 2. Circular Error Probable Matlab Means that 50% of the positions returned calculated have an error lower or equal to the accuracy value. Debt The amount of cash shoftfall determines the borrowing needs which determines the interest expense which determines the amount of debt. It allows the x- and y-coordinates to be correlated and have different variances. http://allsoftwarereviews.com/circular-error/circular-error-probable-cep.php For one-dimensional distributions: Sigma Percentile 0,67 0,5 (CEP) 0,80 0,58 (mean error) 1 0,6827 (rms and std deviation) 1,15 0,75 1,96 0,95 2 0,9545 2,33 0,98 2,57 0,99 3 0,9973 4 If systematic accuracy bias is taken into account, numerical integration of the multivariate normal distribution around an offset circle is required for an exact solution. Circular Error Probable Gps Root Mean Square Error (rms): The square root of the average of the squared error. CEP is not a good measure of accuracy when this distribution behavior is not met. ## In the special case where we assume uncorrelated bivariate normal data with equal variances the Rayleigh estimator does provide true confidence intervals, and it is easy to calculate using spreadsheets. and Bickert, B. (2012). "Estimation of the circular error probability for a Doppler-Beam-Sharpening-Radar-Mode," in EUSAR. 9th European Conference on Synthetic Aperture Radar, pp. 368-371, 23-26 April 2012. To determine the amount of interst, we use an average of the forecasted beginning and ending cash balances. Are circular references bad? Circular Error Probable Calculator The Ethridge (1983) estimator is not based on the assumption of bivariate normality of $$(x,y)$$-coordinates but uses a robust unbiased estimator for the median radius (Hogg, 1967). Rice: When the true center of the coordinates and the POA are not identical, the radial error around the POA in a bivariate uncorrelated normal random variable with equal variances follows Please try the request again. The Ignani (2010) estimate is based on a polynomial approximation for the 50%, 90%, 95%, and 99% quantiles of the Hoyt distribution. navigate here This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any It generalizes to three-dimensional data and can accommodate systematic accuracy bias, but it is limited to the 50% CEP. The first part is to define a cell called jumpstart and the value will be TRUE or FALSE. So for simplicity of forecasting, we just average the beginning and the end and say$3 ($100+$200)/2 * 2%. This approach has the advantage that its calculation is much easier than the exact distribution and does not require special software.

The cumulative distribution function of radial error is equal to the integral of the bivariate normal distribution over an offset disc. For positioning there are 3 variants depending on the number of dimensions being considered: one-dimensional accuracy (used for vertical accuracy), bidimensional accuracy (used for horizontal accuracy) and tridimensional accuracy (combining horizontal Albany, NY: State University of New York Press. For automatic weapons, it is the longest range at which substantial losses are likely to be inflicted on a small area target.

Inventing Accuracy: A Historical Sociology of Nuclear Missile Guidance. Generated Sat, 19 Nov 2016 21:07:51 GMT by s_sg2 (squid/3.5.20) Let's walk through two typical cases. If systematic accuracy bias is taken into account, the Grubbs-Liu estimator has the theoretical advantage over the Grubbs-Pearson estimator that the approximating distribution matches the true distribution not only in mean,


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# Math Help - need help with Solving inequalities using interval notation and grapghing.

1. ## need help with Solving inequalities using interval notation and grapghing.

2. 2x-3/x+1 ≤ 1 (inequality involving quotient)

2. Originally Posted by rsultana

2. 2x-3/x+1 ≤ 1 (inequality involving quotient)
I'll show you how to do the first. The second is done the same way.

$x(x - 1)(x + 2) > 0$

First we want to get everything onto one side of the equation. That's already done here.

Next we want to identify "critical points." This is where the LHS of the equation is 0, or where the denominator is 0. Here we have no denominator, so our critical points are x = -2, 0, 1.

Now we want to break the real line into pieces, given by the critical points. In this case we have 4 intervals:
$( - \infty, -2), (-2, 0), (0, 1), (1, \infty)$

Now we want to test values of x from each interval in the inequality to see which ones give the correct result:
$( - \infty, -2)$ gives $x(x - 1)(x + 2) < 0$ (No!)

$(-2, 0)$ gives $x(x - 1)(x + 2) > 0$ (Yes!)

$(0, 1)$ gives $x(x - 1)(x + 2) < 0$ (No!)

$(1, \infty )$ gives $x(x - 1)(x + 2) > 0$ (Yes!)

Only two of these intervals have x values that work in the inequality. These intervals are the solution set. Thus the answer is $( -2, 0) \cup (1, \infty )$.

The answer to the second problem is $(-1, 4]$. (The 4 is included in the solution set because we have a $\leq$ symbol. Why isn't -1 included in the solution set?)

-Dan

3. Originally Posted by topsquark
I'll show you how to do the first. The second is done the same way.

$x(x - 1)(x + 2) > 0$

First we want to get everything onto one side of the equation. That's already done here.

Next we want to identify "critical points." This is where the LHS of the equation is 0, or where the denominator is 0. Here we have no denominator, so our critical points are x = -2, 0, 1.

Now we want to break the real line into pieces, given by the critical points. In this case we have 4 intervals:
$( - \infty, -2), (-2, 0), (0, 1), (1, \infty)$

Now we want to test values of x from each interval in the inequality to see which ones give the correct result:
$( - \infty, -2)$ gives $x(x - 1)(x + 2) < 0$ (No!)

$(-2, 0)$ gives $x(x - 1)(x + 2) > 0$ (Yes!)

$(0, 1)$ gives $x(x - 1)(x + 2) < 0$ (No!)

$(1, \infty )$ gives $x(x - 1)(x + 2) > 0$ (Yes!)

Only two of these intervals have x values that work in the inequality. These intervals are the solution set. Thus the answer is $( -2, 0) \cup (1, \infty )$.

The answer to the second problem is $(-1, 4]$. (The 4 is included in the solution set because we have a $\leq$ symbol. Why isn't -1 included in the solution set?)

-Dan
i am sorry but i dont get the graphing part of the inequalities.for the second problem i can get to x-4/x+1≤0 but from there on wards i dont know what to do!and i have no idea why -1 is not included.is it because we got the -1 from subtracting it frm the other side?

4. Originally Posted by rsultana
i am sorry but i dont get the graphing part of the inequalities.for the second problem i can get to x-4/x+1≤0 but from there on wards i dont know what to do!and i have no idea why -1 is not included.is it because we got the -1 from subtracting it frm the other side?
Well, I didn't help you with the graphing, all I gave you was what intervals of the real line satisfy the inequality.

The way I learned how to graph the first problem (there are at least two conventions being used)
$( -2, 0) \cup (1, \infty )$
would be to put an open circle on x = -2, x = 0, and x = 1, then draw a line connecting x = -2 and x = 0 to indicate that these points are not part of the solution set, then draw a line to the right starting at x = 1 and continuing to $\infty$. (I put an arrow on the line from x = 1.)

For the second problem:
$(-1, 4]$
I put an open circle on -1 and a closed circle on 4 (to indicate that x = 4 is part of the solution) and connect x = -1 and x = 4.

Why can't x = -1 in the second solution? First of all, PLEASE use parenthesis!!
$x-4/x+1 = x - (4/x) + 1$

What you wanted was "(x - 4)/(x + 1)"

Anyway, plug x = -1 into the original inequality. Is this possible? What about x = 4?

-Dan

5. Originally Posted by topsquark
Well, I didn't help you with the graphing, all I gave you was what intervals of the real line satisfy the inequality.

The way I learned how to graph the first problem (there are at least two conventions being used)
$( -2, 0) \cup (1, \infty )$
would be to put an open circle on x = -2, x = 0, and x = 1, then draw a line connecting x = -2 and x = 0 to indicate that these points are not part of the solution set, then draw a line to the right starting at x = 1 and continuing to $\infty$. (I put an arrow on the line from x = 1.)

For the second problem:
$(-1, 4]$
I put an open circle on -1 and a closed circle on 4 (to indicate that x = 4 is part of the solution) and connect x = -1 and x = 4.

Why can't x = -1 in the second solution? First of all, PLEASE use parenthesis!!
$x-4/x+1 = x - (4/x) + 1$

What you wanted was "(x - 4)/(x + 1)"

Anyway, plug x = -1 into the original inequality. Is this possible? What about x = 4?

-Dan
if we plug in x=-1 then we can get an undefined problem but we get 0 if we plug in x=4


text:
## happykiddo 3 years ago solve the equation by finding the square root x^2-9=0 i got x=4 + or - √4

1. anonymous

how did you get 4...

2. anonymous

i think you meant to type 3

3. anonymous

|dw:1360110109271:dw| to solve by square root; get the x^2 term by itself and take the square root of each side.

4. happykiddo

i posted the answer for the wrong question. sorry. would you be willing to help again heres the my actual question 2x^2-64=0 and i got x=4 + or - √4 for this question

5. anonymous

$2x ^{2}=64$ $x ^{2}=32$ $x=+-\sqrt{32}$

6. happykiddo

could i simplify √32 ?

7. happykiddo

like x=-6+- √27 can be simplified into x=-6+- 3√3

8. anonymous

of course it would be $\sqrt{16\times2}$ so $\sqrt{16}\times \sqrt{2}$=$4\sqrt{2}$

9. anonymous

exact @happykiddo

10. happykiddo

thanks alot :)

11. anonymous

u're welcome!


text:
# Is this operator continuous?

Let $$I=[0,1]$$ and $$E$$ a Banach space. We note by $$X:=\mathcal {C}(I,E),$$ the space of all continuous functions from $$I$$ to $$E$$, with $$\left \| x \right \|_X=\sup_{t\in I }\left \| x(t) \right \|_E$$.

Let $$f:I\times E\rightarrow E$$ a function such that:

• For each continuous $$x\in X$$, we have $$f(.,x(.))$$ is Pettis integrable on $$I$$,

• for every $$t \in I,\:\: f_t: E \rightarrow E,\:u \mapsto f_t(u):=f(t,u) \text{ is continuous}.$$

Let $$T: X \rightarrow X,\:x \mapsto T(x)(t):=\int_{0}^{t}f(s,x(s)) ds$$

Claim: $$T$$ is continuous.

This is how I tried to solve this:

For $$t\in I,\:f_t$$ is continuous, that is,

for each $$u\in E$$, $$\forall \epsilon>0 , \exists \eta_{t,u,\epsilon}>0 \text{ such that } \forall v\in E$$ $$\left \|u-v \right \| \leq \eta_{t,u,\epsilon} \Rightarrow \left \| f(t,u)-f(t,v) \right \| < \epsilon$$

Now, let $$t\in I$$, $$\epsilon >0$$ , and $$x\in X$$. Let $$y\in X$$ such that $$\left \| x-y \right \|_X\leq \eta_{t,x(t),\epsilon}\;,$$

i.e. $$\forall s\in I,\:\left \| x(s)-y(s) \right \|_{E\times E}\leq \eta_{t,x(t),\epsilon}\;,$$ in particular, $$\left \| x(t)-y(t) \right \|_{E\times E}\leq \eta_{t,x(t),\epsilon}\;.$$

Hence, $$\left \| f(t,x(t))-f(t,y(t)) \right \| < \epsilon \quad(*)$$

So, $$\begin{matrix} \left \| T(x)(t)-T(y)(t) \right \| & = &\left \| \int_{0}^{t} f(s,x(s))-f(s,y(s)) ds\right \| \\ & \leq & \int_{0}^{t} \left \| f(s,x(s))-f(s,y(s)) ds\right \| \quad(**)\\ \end{matrix}$$

unfortunately, I can't use $$(*)$$ in $$(**)$$ because it $$(*)$$ not uniformaly on $$t$$.

Is our claim true? why?

If not, what is the condition on $$f_t$$ that you suggest instead of continuity?

I do not have a counterexample, but a strong feeling that the conjecture is false, based on the following positive proof.

If you require slightly more, namely Lebesgue integrability of $$t\mapsto f(t,x(t))$$ for every $$x\in Z=L_\infty([0,1],E)$$, then it is well-known that the superposition operator $$F(x)(t)=f(t,x(t))$$ is “automatically” continuous from $$Z$$ into $$Y=L_1([0,1],E)$$ (because $$f$$ is a Carathéodory function, and $$Y$$ is a regular ideal space), see e.g. Theorem 5.2.1 in M. Väth, Ideal spaces, Springer, 1997.

This implies the continuity of $$T\colon Z\to X$$, because if $$x_n\to x$$ in $$X$$ then $$x_n\to x$$ in $$Z$$, and so

$$\lVert T(x_n)-T(x)\rVert_X\le\lVert F(x_n)-F(x)\rVert_Y\to0\text.$$

I am not sure whether the automatic continuity of $$F$$ holds also in case $$Z=C([0,1],E)$$, but I recall that such a result was cited in case $$E=\mathbb R$$, and I was unable to access it. However, probably the above mentioned proof can be modified straightforwardly for that case.

Anyway, the proof in the “simpler” case $$Z=L_\infty([0,1],E)$$ already depends on a subtle application of Vitali's convergence theorem (much more subtle than the straightforward attempt in the question), and it is hard to imagine that there is any proof which comes without using some sort of equi-integrability of the sequence $$F(x_n)$$. Since the latter is exactly what you lack for Pettis integrability, I would be very surprised if the assertion would be true.

• Thank you for your answer, and for the reference. I hhave some remarks here: 1) Since $\left \| T \right \|_Z\leq \left \| . \right \|_X$, why $\left \| T(x_n)-T(x) \right \|_X\leq \left \| F(x_n)-F(x) \right \|_Z$ ? 2)Do you suggest that I suppose that $t \mapsto f(t,x(t))$ is Pettis integrable for every $x\in Z$ instead of $x\in X$? – Motaka 2 days ago
• 1. For every $t\in[0,1]$ there holds $\lVert T(x_n)(t)-T(x)\rVert_E\le\int_0^t\lVert F(x_n)(s)-F(x)(s)\rVert_E\,ds\le\lVert F(x_n)-F(x)\rVert_{L_1([0,1],E)}$, hence the right-hand side is a bound for the maximum over all $t\in[0,1]$. – Martin Väth 2 days ago
• 2. No, I assume that for the Pettis integral there is no result at all about automatic continuity, even under the stronger hypothesis you mention. – Martin Väth 2 days ago
• A misunderstanding made me think that $F$ and $T$ are the same up to the starting space. For $2$, unfortunately, the conditions that I have are those mentioned in the question. – Motaka 2 days ago


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# CBSE 10th Maths: Real Numbers

NCERT Exercise Solution: 1.1 to 1.4
1. #### Exercise 1.1 | 5 questions

Finding HCF using Euclid's Division Algorithm, word problems on HCF, expressing positive integers, their squares and cubes in the form 6q + r, 3q + r, and 9q + r.

2. #### Exercise 1.2 | 7 questions

Fundamental theorem of arithmetic, prime factorization is unique, finding LCM and HCF by prime factorization, product of 2 numbers is equal to the product of LCM and HCF of the 2 numbers and word problems in LCM and HCF.

3. #### Exercise 1.3 | 3 Questions

Proving that numbers such as √5 are irrational. The method is by assuming the contrary and negating the assumption. Further proving that product of a rational and irrational is irrational and sum of a rational and an irrational number is irrational.

Coming Soon


text:
# A non-stable bipartite degree sequence on which the swap Markov chain is rapidly mixing

### Abstract

The swap Markov chain on a bipartite degree sequence $d$ operates on realizations of $d$ by taking two-two vertices from each color class uniformly, and if the four vertices induce exactly two disjoint edges, replaces them with the two edges induced in the complement. A bipartite degree sequence is stable, if reducing the degrees of one vertex from each color class changes the number of realizations by at most a factor of a polynomial of the number of vertices. Recently, we have shown that the swap Markov chain is rapidly mixing on the realizations of stable (bipartite) degree sequences.

Date
Event
Combinatorics seminar
Location
Alfréd Rényi Institute, Budapest
###### Mathematician

My research interests include Graph Theory, Computational Geometry, and Algorithms


text:
How to convert from Object space into World space (exercise from 3D Math Primer book)

This is not homework. I'm self studying Computer Graphics, using the book "3D Math Primer for Graphics and Game Development" (2nd edition). So there is the exercise 6 in chapter 3 which I can't figure out how to solve exactly. It goes like this:

"Assume that the robot is at the position (1, 10, 3), and her right, up, and forward vectors expressed in upright space are [0.866, 0, −0.500], [0, 1, 0], and [0.500, 0, 0.866], respectively. (Note that these vectors form an orthonormal basis.) The following points are expressed in object space. Calculate the coordinates for these points in upright and world space. (a) (−1, 2, 0) (b) (1, 2, 0) ..."

Now I know that in order to convert from Object space to world space I have to first convert into inertial/upright space by performing a rotation and then into world space by performing a translation. So I assume that that the point (1,10,3) is expressed in world space coordinates. What I don't understand is which one is the matrix I have to use to do the rotation on point (-1,2,0)? I understand conceptually what is going on but I don't know which operations to do here. Help is greatly needed and will be appreciated. Thanks in advance.

• Also, I'm not sure I understand what the right, up and forward vectors represent (I know it's linked to x, y, z axes but don't see exactly how these fit). – Nik-Lz Feb 7 at 11:19

Let me try to translate the problem statement in a mathematical way so that it can be understood more easily.

Assume that the robot is at the position (1, 10, 3)

Assume there are a set of basis $$\mathbf{e}_i$$ and origin $$\mathbf{O}$$, then the robot position $$\mathbf{P}$$ expressed in this basis is the origin plus a linear combination of the basis and the coefficients $$\mathbf{p}=(1,10,3)$$: $$\mathbf{P}=p_i \mathbf{e}_i + \mathbf{O}$$

and her right, up, and forward vectors expressed in upright space are [0.866, 0, −0.500], [0, 1, 0], and [0.500, 0, 0.866], respectively.

Now we have another set of basis denoted as $$\mathbf{e}'_i$$ attached at $$\mathbf{P}$$, and the relation between these two basis $$\mathbf{e}$$ and $$\mathbf{e}'$$ are also expressed as linear combinations. That is, denote $$\mathbf{m}_0=(0.866,0,-0.5)$$,$$\mathbf{m}_1=(0,1,0)$$, and $$\mathbf{m}_2=(0.5,0,0.866)$$, then $$\mathbf{e}'_i=m_{ij} \mathbf{e}_j$$

Calculate the coordinates for these points in upright and world space. (a) (−1, 2, 0) (b) (1, 2, 0) ...

The given coordinates are in object space and the problem asks us to compute their representations in frames $$\{\mathbf{e}_i,\mathbf{P}\}$$ and $$\{\mathbf{e}_i,\mathbf{O}\}$$ respectively.

Take $$\mathbf{x}=(-1,2,0)$$ as an example, we want to find the coordinates $$\mathbf{l}$$ in upright space: \begin{align} x_i \mathbf{e}'_i + \mathbf{P} &= l_i \mathbf{e}_i + \mathbf{P}\\ x_j m_{ji} \mathbf{e}_i &= l_i \mathbf{e}_i \end{align} Written in matrix form: $$\mathbf{M}^T \mathbf{x}=\mathbf{l}$$.

Next we find the coordinates $$\mathbf{w}$$ in world space: \begin{align} x_i \mathbf{e}'_i + \mathbf{P} &= w_i \mathbf{e}_i + \mathbf{O} \\ x_i \mathbf{e}'_i + p_i\mathbf{e}_i &= w_i \mathbf{e}_i \\ x_j m_{ji} \mathbf{e}_i + p_i\mathbf{e}_i &= w_i \mathbf{e}_i \\ x_j m_{ji} + p_i &= w_i \end{align} Written in vector form: $$\mathbf{M}^T \mathbf{x} + \mathbf{p} = \mathbf{w}$$

Your idea is generally correct. And the "upright" matrix can be constructed by considering the relation ($$m_{ij}$$) between the two basis, as shown above.


text:
Hi there,

How many rad or Sv can this IC withstand before it performance degrade?

Regards,

Carmen.

Parents
• Hi,

I got a feedback now from the responsible Apps Engr.

The environment you have mentioned is comparable to Low Dose Radiation (LDR) environment.
We do not have exact data for ADA4800 but we have data for ADA4084-2S and this should have similar data but not exact.


text:
# Euler–Lagrange equation (changing variable)

Create the Euler–Lagrange equation for the following questions (if it's necessary change the variables).

$$\tag{1}\int_{y_1}^{y_2}\dfrac{x'^2}{\sqrt{x'^2+x^2}}\,\mathrm{d}y$$

$$\tag{2}\int_{x_1}^{x_2}y^{3/2}\,\mathrm{d}s$$

$$\tag{3}\int\dfrac{y\cdot{y'}}{1+yy'}\,\mathrm{d}x$$

I don't have an idea about $(1)$ and $(3)$. But here it's what I have tried for $(2)$.

2) $$\int_{x_1}^{x_2}y^{3/2}\,\mathrm{d}s = \int_{x_1}^{x_2}y^{3/2}(1+y'^2)^{1/2} = \int_{y_1}^{y_2}(1+x'^2)(y^{3/2})\,\mathrm{d}y$$

So our Euler equation is:

$$\dfrac{\mathrm{d}}{\mathrm{d}y}\left(\dfrac{\partial F}{\partial x'}\right) - \dfrac{\partial F}{\partial x} = 0$$

Then I have to find $Y'$ or $X'$. But I did not take a differential equations course yet, we use Beltrami identity to calculate the extremum points.

-

We know that the functional

$$I[y]=\int_{x_1}^{x_2} f(x,y(x),y'(x))\,\mathrm{d}x$$

is extremized only if $y$ satisfies the Euler–Lagrange equation

$$\frac{\partial{f}}{\partial{y}}-\frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial{f}}{\partial{y'}}=0.$$

Now, if $f$ is independent of $x$, then the Euler–Lagrange equation reduces to the Beltrami identity

$$f-y'\frac{\partial{f}}{\partial{y'}}=C=\text{constant}.$$

Note that the names of the variables are immaterial. Henceforth, we use the same notation as above.

In the first case, we have $f(x,y,y')=\dfrac{y'^2}{\sqrt{y^2+y'^2}}$, which is independent of $x$. Hence we can use the Beltrami identity. Differentiation gives

$$\frac{\partial{f}}{\partial{y'}}=\frac{2y^2y'+y'^3}{\left(y^2+y'^2\right)^{3/2}}\implies\frac{y'^2}{\sqrt{y^2+y'^2}}-\frac{2y^2y'^2+y'^4}{\left(y^2+y'^2\right)^{3/2}}=C.$$

Rewriting the first term gives

$$\frac{y^2y'^2+y'^4}{\left(y^2+y'^2\right)^{3/2}}-\frac{2y^2y'^2+y'^4}{\left(y^2+y'^2\right)^{3/2}}=C\implies-\frac{y^2y'^2}{\left(y^2+y'^2\right)^{3/2}}=C,$$

which is a first-order non-linear ordinary differential equation. In total, two constants will be obtained.

Of course, we could use the Euler–Lagrange equation instead, but that would be much more complicated; just consider the derivatives:

$$\frac{\partial{f}}{\partial{y}}=-\frac{yy'^2}{\left(y^2+y'^2\right)^{3/2}},\,\frac{\partial{f}}{\partial{y'}}=\frac{2y^2y'+y'^3}{\left(y^2+y'^2\right)^{3/2}},\,\frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial{f}}{\partial{y'}}=\frac{y(2y^2-y'^2)(yy''-y'^2)}{\left(y^2+y'^2\right)^{5/2}}.$$

The second and thrid cases can be treated similarly; both integrands are independent of $x$. However, in the second case, note that $\mathrm{d}s=\sqrt{1+y'^2}\,\mathrm{d}x$, so $f(x,y,y')=y^{3/2}\sqrt{1+y'^2}$; this expression is independent of $x$, as it should be.

-
you explanation is perfect but when I'm trying to derivate the functional,I can't get the same answer as you. – Erbil Jun 17 '13 at 14:11
$$\dfrac {\partial F} {\partial y'}=\dfrac {2y'\sqrt {y^{2}+y'^{2}}-2y'y'^{2}} {\left( y^{2}+y'^{2}\right)^{3/2} }$$ – Erbil Jun 17 '13 at 14:36
@Erbil It follows from the quotient rule: $$\frac{\partial{f}}{\partial{y'}}=\frac{2y'\sqrt{y^2+y'^2}-y'^3\Big/\sqrt{y^2+y'^2}}{y^2+y'^2}=\frac{2y^2y'+y'^3}{\left(y^2+y'^2\right)^{3/2}}.$$ – Librecoin Jun 17 '13 at 18:25
I think I have used the same rule,and I have also take a care about partial derivative.But where did you get +,and where did the 2 coefficient go on y'3? edit : There is more than these.I can't also understand where the (y^2+y'^2)^1/2 gone :( – Erbil Jun 17 '13 at 21:11
@Erbil Multiply the numerator and the denominator in the first fraction by $\sqrt{y^2+y'^2}$, and the equality follows. – Librecoin Jun 17 '13 at 21:39


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# What are the Lewis dot formulae for sulfuric and nitric acids?

Dec 28, 2015

Both sulfuric and nitric acids are NEUTRAL species. Lewis dot formulae should (and do) reflect this neutrality.

#### Explanation:

Typically, we would display the sulfur in sulfuric acid as a neutral species, with 2 $S = O$ bonds. Thus, we would get the Lewis structure: $\left(H O\right) S {\left(= O\right)}_{2} \left(O H\right)$. Alternatively, we could represent charge separation: $\left(H O\right) {S}^{2 +} {\left(- {O}^{-}\right)}_{2} \left(O H\right)$, which is arguably a better representation.

On the other hand, nitric acid MUST display a quaternized nitrogen (i.e. formally positive), and a formally negative oxygen charge: $H O - {N}^{+} \left(= O\right) {O}^{-}$.

All of these Lewis structures are representations of actual NEUTRAL and ISOLABLE molecules. The Lewis structures that I have tried to represent are conceptual, and may or may not represent chemical reality.

Neither Lewis representation will indicate how or why sulfuric acid is a stronger acid than nitric acid. This is an experimental phenomenon, and is not dependent our Lewis representations. In fact, sulfuric acid is a stronger acid than nitric acid, and will protonate nitric acid to give the $N {O}_{2}^{+}$ cation:

${H}_{2} S {O}_{4} + H N {O}_{3} \rightarrow N {O}_{2}^{+} + H S {O}_{4}^{-} + {H}_{2} O$

I've forgotten what you call the $N {O}_{2}^{+}$ cation! It might come to me presently. Ah, it's the nitronium ion.