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# Collective responses in electrical activities of neurons under field coupling

Scientific Reportsvolume 8, Article number: 1349 (2018) | Download Citation

## Abstract

Synapse coupling can benefit signal exchange between neurons and information encoding for neurons, and the collective behaviors such as synchronization and pattern selection in neuronal network are often discussed under chemical or electric synapse coupling. Electromagnetic induction is considered at molecular level when ion currents flow across the membrane and the ion concentration is fluctuated. Magnetic flux describes the effect of time-varying electromagnetic field, and memristor bridges the membrane potential and magnetic flux according to the dimensionalization requirement. Indeed, field coupling can contribute to the signal exchange between neurons by triggering superposition of electric field when synapse coupling is not available. A chain network is designed to investigate the modulation of field coupling on the collective behaviors in neuronal network connected by electric synapse between adjacent neurons. In the chain network, the contribution of field coupling from each neuron is described by introducing appropriate weight dependent on the position distance between two neurons. Statistical factor of synchronization is calculated by changing the external stimulus and weight of field coupling. It is found that the synchronization degree is dependent on the coupling intensity and weight, the synchronization, pattern selection of network connected with gap junction can be modulated by field coupling.

## Introduction

Neurons are the basic structural and functional units of nervous system. The information processing and encoding are approached by exchange of neurotransmitter via chemical and electrical action, and the signal receiving is dependent on synapses between neurons1,2,3,4,5,6,7. For example, Qin et al.5 discussed the biological function of autapse connection to neurons and confirmed that autapse driving can enhance self-adaption of neurons. The biological nervous system is usually very complex and hierarchical, and it contains very complex inner structures, such as multiple subsystems. Therefore, the research in internal competition and cooperation behavior of complex system has to resort to the sampling and system analysis. The related studies have very great significance on understanding the evolution of collective behaviors of complex systems, and it is also helpful to predict the dynamical behavior of control systems. For an isolated neuron, geometry and anatomical structure often account for biological function, for example, autapse formation could be associated with injury in axon thus auxiliary loops are developed to propagate the blocked signals8. Moreover, the impact of different topology networks in various neuron models has been widely investigated by using bifurcation analysis. Indeed, coupling strength9,10, time delay11, spatial in homogeneity stimulus12, and noise13,14 can affect the synchronization and stability of network. The network is composed of many nodes, the statistical analysis based on mean field theory and nonlinear analysis from sampled time series of observable variables are often used to detect the synchronization stability and pattern selection15,16. However, it could be difficult for generic sampling and statistical analysis when network is composed of many sub-networks, but pattern selection could be effective to understand the transition in collective behaviors. On the other hand, the use of speckle graph dynamics is much helpful to study the group discharge behaviors of neuronal networks, thus the spiral wave characteristics of spatial distribution of electrical activity in the cerebral cortex17 can be understood. The spiral wave is self-sustained, and the developed spiral waves in cortex can change the collective behaviors in electrical activities as a stable pacemaker. The potential mechanism for emergence of spiral wave and its dynamics are discussed in ref.18, which local and diffusive poisoning in ion channels19,20,21 can generate defects to trigger continuous waves and further occurrence of spiral waves after symmetrical breaking22 in the profile. For example, Hu et al.22 confirmed that symmetry breaking of target wave in the media accounts for emergence of multiarmed spiral wave, which can be developed from a group of spiral waves with single arm under appropriate condition, thus the potential formation mechanism of multiarmed spiral wave in the media is explained. Gong et al23. discussed the influence of time delay and channel blocking on multiple coherence resonance in network composed of Hodgkin-Huxley neurons. Qin et al.24 explained the defect mechanism for autapse driving in network by using negative feedback and the wave propagation was also discussed. Xiao et al.25 reported the dynamics in spatiotemporal patterns in network by changing the excitability. Xu et al.26 explained the new mechanism of spatial coherence resonance by imposing noise and periodical pacing in different local areas in the network, and it is found that local forcing-induced wave can be broken in the area driven by noise and spiral waves are formed under appropriate noise intensity in the network. Li et al.27 discussed the selection and breakup of spiral wave in a coupled network driven by Gaussian colored noise. Qian et al.28 analyzed all oscillatory complex networks consisting of non-oscillatory nodes by using the dominant phase-advanced driving method, the oscillation sources and wave propagation paths of the self-sustained target waves in excitable small-world networks were explored explicitly.

In fact, spatial patterns such as spiral wave, target wave and Turing patterns can also be found in Reaction-diffusion systems and neuronal networks as well. Pattern formation in nonlinear complex systems is one of the central problems of the natural, social, and technological sciences. Since Turing firstly proposed reaction-diffusion (RD) theory to model the range of spatial patterns observed in the developing embryo29, RD models have been studied extensively to explain patterns in both epidemiology and ecosystems30. It is found that cross diffusion31, time delay32,33 and functional response34 can induce the appearance of stationary patterns. It is confirmed that the standard multiple-scale analysis and exact Turing regions can be obtained35 to develop kinds of spatial patterns. In fact, spiral wave, target wave or patched invasion can be observed36,37 when the diffusion coefficients are the same (Turing theory does not hold). Based on the data observations, it is revealed that multiple scale spatial patterns38, isolation degree39 and spatial scaling laws40 play important roles on the resilience of the biological systems and thus they have implications on population protection or diseases control. In cardiac tissue and also cortex, the emergence and survival of spiral waves, continuous pulses could disturb signal propagation and information processing, thus normal electrical activities in cortex and wave propagation in cardiac tissue can be perturbed for inducing possible diseases. Therefore, it is important to apply effective schemes40,41 to remove and suppress spiral waves. Based on the driven-synchronization, the rotating electric field (REF) can be utilized to “smooth” a heterogeneous medium and suppress turbulence42. Chen et al.43,44 found that the REF can induce wave emission from heterogeneity in excitable media, which may provide a potential application to utilized the existed defects on cardiac tissue as the second sources to remove pinned spiral waves45.

That is, the effect of electric field on cardiac tissue can often be described by adding gradient terms in the membrane potential of media described by the Reaction-Diffusion system. In the cortex and neuronal network, it is a challenge to deal with this problem that the effect of electromagnetic induction can be understood from physical view. The influence of the biological effects of electromagnetic fields on human health has been investigated, and electromagnetic radiation-induced disease and abnormality in nervous systems (such as memory loss and other symptoms) were also discussed46,47,48,49,50,51,52,53,54,55. On the other hand, the dynamics of neuronal activities in presence of electromagnetic field also became attractive56,57,58. In fact, by further improving the computational models for neuron59,60 and cardiac tissue61,62, the physical effect can be described by using magnetic flux associated with time-varying electromagnetic field. For example, Lv et al.63 proposed a four-variable neuron model developed from the previous Hindmarsh-Rose neuron model59,60 by adding the variable for magnetic flow, which memristor is also used to bridge the coupling between membrane potential and magnetic flux63. Inspired by this model, the transition of electrical activities in neurons induced by electromagnetic radiation has been investigated extensively64,65,66,67,68, particularly, the same scheme was carried out on cardiac tissue, and two death mechanisms61,62 in heart induced by electromagnetic radiation are explained. For a further survey and guidance, readers can find clues in refs69,70 and references therein. In most of the previous works, synapse connection and gap junction are regarded as the main bridge to benefit signal exchange and wave propagation between neurons. In this paper, we argued that field coupling could be another effective way for signal propagation in network because field coupling can realize phase synchronization between neurons71. The potential mechanism is explained that each neuron can contribute to the distribution of electric field, and the electrical activity of each neuron will be affected by other neurons by triggering different electric fields in the network.

## Model and scheme

The Hindmarsh-rose neuron model59,60 mainly highlights the main nonlinear dynamic characteristics of mollusk neurons, the dynamical kinetics can be approached by the ordinary differential equations (ODEs) composed of three variables shown in ref.59. As proposed and discussed in refs63,64, magnetic flux is used to describe the effect of electromagnetic induction72, and memristor is considered to realize coupling between membrane potential and magnetic flux thus the induction field and action potential can be bridged in physical view. The dynamical neuron model is described by

$$\{\begin{array}{l}\frac{dx}{dt}=y-a{x}^{3}+b{x}^{2}-z+I-k\rho (\phi )x\\ \frac{dy}{dt}=c-d{x}^{2}-y\\ \frac{dz}{dt}=r[s(x+1.56)-z]\\ \frac{d\phi }{dt}={k}_{1}x-{k}_{2}\phi \end{array}$$
(1)

where the variable x, y, z represents the membrane potential, slow current for recovery variable, and adaption current, respectively. The variable I defines the external forcing current, and the fourth variable φ describes the magnetic flux across membrane. The ρ(φ) is the memory-conductance of a magnetic flux-controlled memristor and it is approached as ρ(φ) = α + 3βφ2, (φ)x denotes induction current and k is induction coefficient. Further detailed description for parameters can be found in refs63,64. The induction current is calculated as follows

$$i^{\prime} =\frac{dq(\phi )}{dt}=\frac{dq(\phi )}{d\phi }\frac{d\phi }{dt}=\rho (\phi )\frac{d\phi }{dt}=-k\rho (\phi )x$$
(2)

That is, the effect of electromagnetic induction and change of electric field is described by induction current. The memductance of memristor is dependent on the magnetic flux and time-varying induction current is imposed on the neuron. The memristor73 is often used as nonlinear electric device in setting nonlinear circuits and synchronization can be approached by using memristor coupling74,75. It is also effective to describe the effect of electromagnetic induction in neuronal activities. From physical view, each neuron and cell can be thought as an non-uniform charged body because the distribution of charged ions are not distributed in uniform way, and thus complex field distribution can be triggered in intracellular and extracellular space. The effect of electric field could be distinct and the time-varying electromagnetic field plays important role in changing the exchange of charged ions. In biological and nervous systems, contribution to the field distribution from each neuron could be different and shows diversity, therefore, appropriate weight is considered in the neuronal network. For simplicity, in a chain-like network, the position of neuron is marked with subscript “i”, another neuron is marked with subscript “j”, it is supposed that the contribution to field intensity is inversely proportional to the number error as Γ ij  = W/│i − j│, and W is the weight value contributed to the field intensity. As a result, the network can be approached by gap junction coupling and field coupling; the dynamical equations are described by

$$\{\begin{array}{l}\frac{d{x}_{i}}{dt}={y}_{i}-a{{x}_{i}}^{3}+b{{x}_{i}}^{2}-{z}_{i}+I+D({x}_{i+1}+{x}_{i-1}-2{x}_{i})-k\rho ({\phi }_{i}){x}_{i}\\ \frac{d{y}_{i}}{dt}=c-d{{x}_{i}}^{2}-{y}_{i}\\ \frac{d{z}_{i}}{dt}=r[s({x}_{i}+1.56)-{z}_{i}]\\ \frac{d{\phi }_{i}}{dt}={k}_{1}{x}_{i}-{k}_{2}{\phi }_{i}+{D}_{0}({\phi }_{i}-\sum _{\begin{array}{c}j=1\\ i\ne j\end{array}}\frac{W}{|i-j|}{\phi }_{j})\end{array}$$
(3)

The subscript i indicates the position of the neuron in the network, D is the coupling coefficient between adjacent neurons via electric synapse coupling, D0 describes the field interaction between neurons. W represents the intensity of the field effect associated with distance between neurons. According to Eq. (3), increasing the value for intensity of field coupling, positive feedback will enhance the electromagnetic induction effect, and stronger induction current will be induced to decrease the excitability of neurons. The parameters are selected as a = 1.0, b = 3.0, c = 1.0, d = 5.0, s = 4.0, α = 0.1, β = 0.02, k = 0.9, k1 = 0.4, k2 = 0.5. The field coupling is illustrated in Fig. 1.

To discern the collective behavior of neurons, a statistical factor of synchronization24,26,69 is calculated to find synchronization approach based on mean field theory, and it reads as follows

$$F=\frac{{\bf{1}}}{N}\sum _{i=1}^{N}{x}_{i},\quad R=\frac{ < \,{F}^{2} > - < F{ > }^{2}}{\frac{{\bf{1}}}{N}\sum _{i=1}^{N}[ < {x}_{i}^{2} > - < {x}_{i}{ > }^{2}]}$$
(4)

where N is the node number of neuronal network, < * > describes the average of variable over time, for simplicity, a transient period T = 2000 time units will be used for numerical studies. Perfect synchronization is approached when R is close to 1, and non-perfect synchronization is detected together with emergence of appropriate pattern distribution when R is close to zero.

## Numerical results and discussion

In the numerical section, the fourth order Runge-Kutta algorithm is used to approach solutions for the dynamical equations. The chain network is composed of 100 neurons, time step is set as h = 0.01. The initial values are selected random values as (0.3 + ξ, 0.1 + ξ, 5 + ξ, 0, 0.2 + ξ), where ξ is a random number ranged from 0 to 1. Transient period for calculation is about 2000 time units, coupling coefficient D = 0.5. In order to illustrate the effect of field superposition on neuronal discharges, different external stimuli I are applied when the chain neuron W is set to 1. The spatiotemporal evolution of membrane potentials are calculated for dynamical analysis when the node position driven by external stimulus current is changed. Furthermore, the distribution for factors of synchronization on the network is calculated by selecting different external stimuli I and D0. At first, the weight of electric field contribution is fixed at W = 1, the intensity of field coupling is adjusted to observe the development of spatial patterns(spatial distribution of membrane potential), the results are plotted in Fig. 2 and sampled time series for membrane potential of neuron connected to node (i = 80) are shown in Fig. 3.

The excitability of neuron mainly depends on the external forcing current, and larger external stimuli are much helpful to excite neurons and the collective behaviors of network will present oscillating state. When the subthreshold forcing is applied on the network, diversity in initial settings can trigger short transient oscillation and then the network become quiescent state occupied the network completely. By further increasing the intensity of external stimuli, the excitability of neuron is also enhanced, the quiescent neurons are waken up to present spiking and bursting states. Indeed, the field coupling contributes magnetic flux and induction current, as a result, the mode in electrical activities are controlled. As shown in Figs 2d, 3d, further increasing the intensity of field coupling, diversity in magnetic flux and induction currents are induced, as a result, each neuron shows different excitability to trigger appropriate oscillating behaviors in electrical activities.

The results in Fig. 4 confirmed that the field coupling plays important role in regulating the collective behaviors of the network, while slight difference in external stimuli can also enhance the developed state when the intensity of field coupling is highly increased. The sampled time series for some nodes found that bursting behaviors are enhanced under field coupling even stimuli is much weak, the results are shown in Fig. 5.

The change in the discharge period of the neuron is caused by the change of the field interaction between each neuron and the other neurons, that is, D0 is positively correlated with the neuronal discharge period T. It is found that the bursting states show distinct difference and the response intervals are much different. The field coupling can drive all neurons to give appropriate response in time. The distribution for factors of synchronization is calculated in the two-parameter space, and the contribution from field coupling and external forcing is estimated, the results are shown in Fig. 6.

It is confirmed that the synchronization degree is high when external forcing current is weak that excitability is low, as a result, the effect of diversity in initials setting can be suppressed by field coupling. To discern the effect of field coupling, the intensity of electric coupling is removed as D = 0, then external forcing current and field coupling are considered, the results are shown in Figs 7, 8, 9, 10.

It is confirmed that weak excitability can’t support continuous oscillating, while field coupling can enhance the oscillating behaviors in electrical activities. The external stimuli mainly contribute the electrical activities when the intensity of field coupling is weak. Furthermore, the field coupling is enhanced to observe the evolution of collective behaviors, the results are shown in Figs 9, 10.

In the case of high excitability by applying stronger external stimuli, the oscillating behaviors and bursting states can be increased greatly under field coupling. As shown in Fig. 10, multiple modes in electrical activities are induced by further increasing the intensity of field coupling, and all the neurons can give rapid response under field coupling. Extensive numerical results confirmed synchronization can be approached when electric coupling is set as D = 0.5 when diversity in initials setting is removed. As a result, the developed states show some difference when field coupling is triggered in different transient periods. Furthermore, the effect of weight contributed to the supimposed field is discussed, the transition of synchronization factors is estimated by changing the intensity of field coupling, the results are plotted in Fig. 11.

It is found that the weight in field coupling also contributes to the synchronization approach of the network. When the weight for field coupling is much small, the synchronization factors show slight difference and synchronization mainly depends on the electric coupling. The role of weight becomes most important when field coupling is increased in intensity because the contribution from each neuron becomes distinct different than other neurons. When the external stimulus is decreased to I = 1.2, the field coupling can modulate the synchronization behavior greatly (decreases the synchronization) with increasing the intensity of field coupling. However, in case of high excitability and stronger stimulus, the oscillating behaviors and evolution of network mainly depend on the stimuli and the effect of weight shows slight difference on synchronization approach.

In a summary, the evolution of collective behaviors and synchronization degree are dependent on the electric coupling via gap junction, intensity of field coupling, weight for superimposed field, diversity in initials setting. Due to the memory effect of memristor associated with magnetic flux and effect of initials diversity becomes important and distinct. The external forcing current changes the excitability of neuron and the electrical activities of neuronal network can be changed directly. Field coupling can modulate the collective behaviors of network by inducing induction currents with diversity and diversity in excitability of network is triggered. The development and evolution of collective electrical activities of neuronal network are contributed by field coupling, electric coupling when external forcing is fixed. On the other hand, the field coupling between neurons can be considered as external electromagnetic radiation on isolated neuron. Readers can extend this study by setting other networks with different connection types when field coupling is considered.

## Conclusions

An improved neuron model, which the membrane potential and magnetic flux are bridged by using memristor, is used to describe the local kinetics of chain network of neurons. The effect of electromagnetic field is described by magnetic flux. Each neuron is regarded as an charged body and is controlled by the electric field triggered by other neurons. A weight value is introduced to discern the contribution to superimposed field from each neuron, field coupling between neurons is described by exchange of magnetic flux. It is found that field coupling between neurons can change the magnetic flux and induction current, and then the excitability of neurons are changed to modulate the collective behaviors of electrical activities in neuronal network. It could give instructive clues to understand the signal encoding and exchange when synapse coupling is suppressed.

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## References

1. 1.

Pfrieger, F. Roles of glial cells in synapse development, Cell Molec. Life Sci 66, 2037 (2009).

2. 2.

Liu, Q., Xu, Q. & Kang, J. et al. Astrocyte activation of presynaptic metabotropic glutamate receptors modulates hippocampal inhibitory synaptic transmission. Neuron Glia Biol 1, 307 (2004).

3. 3.

Lichtman, J. & Colman, H. Synapse elimination and indelible memory. Neuron 25, 269–78 (2000).

4. 4.

Bi, G. & Poo, M. Synaptic modifications in cultured hippocampal neurons: dependence on spike timing, synaptic strength, and postsynaptic cell type. J Neurosci 18, 10464–72 (1998).

5. 5.

Qin, H. X., Ma, J. & Jin, W. Y. et al. Dynamics of electric activities in neuron and neurons of network induced by autapses. Sci China Technol Sci 57, 936–946 (2014).

6. 6.

Wang, H., Wang, L. & Yu, L. et al. Response of Morris-Lecar neurons to various stimuli. Phys Rev E 83, 021915 (2011).

7. 7.

Yilmaz, E., Uzuntarla, M. & Ozer, M. et al. Stochastic resonance in hybrid scale-free neuronal networks. Physica A 392, 5735–5741 (2013).

8. 8.

Wang, C. N., Guo, S. L. & Xu, Y. et al. Formation of autapse connected to neuron and its biological function. Complexity 2017, 543673 (2017).

9. 9.

Shi, X. & Xi, W. Synchronization and rhythm dynamics of a neuronal network consisting of mixed bursting neurons with hybrid synapses. Int J Mod Phys B 30(16), 1650091 (2016).

10. 10.

Xu, Y., Jin, W. & Ma, J. Emergence and robustness of target waves in a neuronal network. Int J Mod Phys B 29(23), 1550164 (2015).

11. 11.

Wang, Q., Perc, M. & Duan, Z. et al. Spatial coherence resonance in delayed Hodgkin-Huxley neuronal neuronal networks. Int J Mod Phys B 24(9), 1201–1213 (2010).

12. 12.

Wang, Q., Perc, M. & Duan, Z. et al. Delay-enhanced coherence of spiral waves in noisy Hodgkin–Huxley neuronal networks. Phys Lett A 372(35), 5681–5687 (2008).

13. 13.

Jin, W., Lin, Q., Wang, A. & Wang, C. Computer simulation of noise effects of the neighborhood of stimulus threshold for a mathematical model of homeostatic regulation of sleep-wake cycles. Complexity 2017, 4797545 (2017).

14. 14.

Yilmaz, E., Baysal, V. & Perc, M. et al. Enhancement of pacemaker induced stochastic resonance by an autapse in a scale-free neuronal network. Sci China Technol Sci 59, 364–370 (2016).

15. 15.

Ma, J., Xu, Y. & Ren, G. D. et al. Prediction for breakup of spiral wave in a regular neuronal network. Nonlinear Dyn 84, 497–509 (2017).

16. 16.

Song, X. L., Wang, C. N. & Ma, J. et al. Collapse of ordered spatial pattern in neuronal network. Physica A 451, 95–112 (2016).

17. 17.

Huang, X., Xu, W. & Liang, J. et al. Spiral wave dynamics in neocortex. Neuron 68, 978–990 (2010).

18. 18.

Wu, X. Y. & Ma, J. The Formation Mechanism of Defects, Spiral Wave in the Network of Neurons. PLoS One 8, e55403 (2013).

19. 19.

Wu, X., Ma, J. & Li, F. et al. Development of spiral wave in a regular network of excitatory neurons due to stochastic poisoning of ion channels. Commun Nonlinear Sci Numer Simulat 18, 3350–3364 (2013).

20. 20.

Ma, J., Huang, L. & Tang, J. et al. Spiral wave death, breakup induced by ion channel poisoning on regular Hodgkin-Huxley neuronal networks. Commun Nonlinear Sci Numer Simulat 17, 4281–4293 (2012).

21. 21.

Huang, L., Ma, J. & Tang, J. et al. Transition of ordered waves in neuronal network induced by diffusive poisoning of ion channels. J Biol Syst 21, 1350002 (2013).

22. 22.

Hu, B., Ma, J. & Tang, J. Selection of Multiarmed Spiral Waves in a Regular Network of Neurons. Plos One 8(7), e69251 (2013).

23. 23.

Gong, Y., Hao, Y. & Lin, X. et al. Influence of time delay and channel blocking on multiple coherence resonance in Hodgkin-Huxley neuron networks, Biosyst 106(2–3), 76–81 (2011).

24. 24.

Qin, H. X., Wu, Y. & Wang, C. N. et al. Emitting waves from defects in network with autapses. Commun Nonlinear Sci Numer Simulat 23, 164–174 (2015).

25. 25.

Xiao, W. W., Gu, H. & Liu, M. Spatiotemporal dynamics in a network composed of neurons with different excitabilities and excitatory coupling. Sci China Technol Sci 59, 1943–1952 (2016).

26. 26.

Xu, Y., Wang, C. N. & Lv, M. et al. Local pacing, noise induced ordered wave in a 2D lattice of neurons. Neurocomputing 207, 398–407 (2016).

27. 27.

Li, F. & Ma, J. Selection of spiral wave in the coupled network under Gaussion colored noise. Int J Mod Phys B 27(21), 1350115 (2013).

28. 28.

Qian, Y., Huang, X. D. & Hu, G. et al. Structure and control of self-sustained target waves in excitable small-world networks. Phys Rev E 81, 036101 (2010).

29. 29.

Turing, A. M. On the chemical basis of morphogenesis, Philos. Trans R Soc Lond B 237, 37–72 (1952).

30. 30.

Klausmeier, C. A. Regular and irregular patterns in semiarid vegetation. Science 284, 1826–1828 (1999).

31. 31.

Sun, G. Q., Jin, Z. & Li, L. et al. Spatial patterns of a predator-prey model with cross diffusion. Nonlinear Dyn. 69, 1631–1638 (2012).

32. 32.

Sun, G. Q., Wang, S. L. & Ren, Q. et al. Effects of time delay and space on herbivore dynamics: linking inducible defenses of plants to herbivore outbreak. Sci Rep 5, 11246 (2015).

33. 33.

Li, L., Jin, Z. & Li, J. Periodic solutions in a herbivore-plant system with time delay and spatial diffusion, Appl Math. Mod 40, 4765–4777 (2016).

34. 34.

Alonso, D., Bartumeus, F. & Catalan, J. Mutual interference between predators can give rise to Turing spatial patterns. Ecology 83, 28–34 (2002).

35. 35.

Sun, G. Q., Wang, C. H. & Wu, Z. Y. Pattern dynamics of a Gierer-Meinhardt model with spatial effects. Nonlinear Dyn 88, 1385–1396 (2017).

36. 36.

Sun, G. Q., Jin, Z. & Li, L. et al. Self-organized wave pattern in a predator-prey model. Nonlinear Dyn 60, 265–275 (2010).

37. 37.

Sun, G. Q. Mathematical modeling of population dynamics with Allee effect. Nonlinear Dyn 85, 1–12 (2016).

38. 38.

Liu, Q. X., Herman, P. M. J. & Mooij, W. M. et al. Pattern formation at multiple spatial scales drives the resilience of mussel bed ecosystems. Nature Commun 5, 5234 (2014).

39. 39.

Sun, G. Q., Wu, Z. Y. & Jin, Z. et al. Influence of isolation degree of spatial patterns on persistence of populations. Nonlinear Dyn 83, 811–819 (2016).

40. 40.

Ritchie, M. E. & Olff, H. Spatial scaling laws yield a synthetic theory of biodiversity. Nature 400, 557–560 (1999).

41. 41.

Yuan, G., Bao, X. & Yang, S. et al. Dynamics of pulses and spiral waves in excitable media with an anomalous diffusion. Int J Mod Phys B 30(20), 1650127 (2016).

42. 42.

Ma, J., Ying, H. P. & Li, Y. L. Suppression of spiral waves using intermittent local electric shock. Chinese Phys 16, 955–961 (2007).

43. 43.

Chen, J. X., Mao, J. W. & Hu, B. et al. Suppression of spirals and turbulence in inhomogeneous excitable media. Phys Rev E 79, 066209 (2009).

44. 44.

Chen, J. X., Peng, L. & Ma, J. et al. Liberation of a pinned spiral wave by a rotating electric pulse. EPL 107, 38001 (2014).

45. 45.

Chen, J. X., Zhang, H. & Qiao, L. Y. et al. Interaction of excitable waves emitted from two defects by pulsed electric fields. Commun Nonlinear Sci Numer Simulat 54, 202–209 (2018).

46. 46.

Lacy-Hulbert, A., Metcalfe, J. C. & Hesketh, R. Biological response to electromagnetic fields. Faseb J 12, 395–420 (1998).

47. 47.

Carpenter, D. O. Human disease resulting from exposure to electromagnetic fields. Rev Environ Heal 28, 159–172 (2017).

48. 48.

Mejia-Arangure, J. M., Fajardo-Gutierrez, A. & Perez-Saldivar, M. L. et al. Magnetic fields and acute leukemia in children with Down Syndrome. Epidemiol 18, 158–161 (2007).

49. 49.

Foliart, D. E., Pollock, B. H. & Mezei, G. et al. Magnetic field exposure and long-term survival among children with leukaemia. Br J Cancer 94, 161–164 (2006).

50. 50.

Garcia, A. M., Sisternas, A. & Hoyos, S. P. Occupational exposure to extremely low frequency electric and magnetic fields and Alzheimer disease: a meta-analysis. Int J Epidemiol 37, 329–340 (2008).

51. 51.

Feychting, M., Jonsson, F. & Pedersen, N. L. et al. Occupational magnetic field exposure and neurodegenerative disease. Epidemiol 14, 413–419 (2003).

52. 52.

Burdak-Rothkamm, S., Rothkamm, K. & Folkard, M. et al. DNA and chromosomal damage in response to intermittent extremely low-frequency magnetic fields. Mut Res 672, 82–89 (2009).

53. 53.

Salehi, I., Sani, K. G. & Zamani, A. Exposure of rats to extremely low-frequency electromagnetic fields (ELF-EMF) alters cytokines production, Electromag. Biol Med 32, 1–8 (2012).

54. 54.

Gultekin, D. H. & Moeller, L. NMR imaging of cell phone radiation absorption in brain tissue. PNAS 110, 58–63 (2013).

55. 55.

Clark, V. P. & Parasuraman, R. Neuroenhancement: Enhancing brain and mind in health and in disease. NeuroImage 85, 889–894 (2014).

56. 56.

Ren, G., Xu, Y. & Wang, C. Synchronization behavior of coupled neuron circuits composed of memristors. Nonlinear Dyn 88, 893–901 (2017).

57. 57.

Ma, J., Wu, F. & Wang, C. Synchronization behaviors of coupled neurons under electromagnetic radiation. Int J Mod Phys B 31(2), 1650251 (2017).

58. 58.

Yu, K., Wang, J. & Deng, B. et al. Synchronization of neuron population subject to steady DC electric field induced by magnetic stimulation. Cogn Neurodyn 7(3), 237–252 (2013).

59. 59.

Hindmarsh, J. L. & Rose, R. M. A Model of Neuronal Bursting Using Three Coupled First Order. Differential Equations, Proc Roy Soc London B 221(1222), 87–102 (1984).

60. 60.

Izhikevich, E. M. Which model to use for cortical spiking neurons? IEEE Tr Neural Netw15 5, 1063–1070 (2004).

61. 61.

Wu, F. Q., Wang, C. N. & Xu, Y. et al. Model of electrical activity in cardiac tissue under electromagnetic induction. Sci Rep 6, 28 (2016).

62. 62.

Ma, J., Wu, F. Q. & Hayat, T. et al. Electromagnetic induction and radiation-induced abnormality of wave propagation in excitable media. Physica A 486, 508–516 (2017).

63. 63.

Lv, M., Wang, C. & Ren, G. et al. Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn 85, 1479–1490 (2016).

64. 64.

Lv, M. & Ma, J. Multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Neurocomputing 205, 375–381 (2016).

65. 65.

Xu, Y., Ying, H. P. & Jia, Y. et al. Autaptic regulation of electrical activities in neuron under electromagnetic induction. Sci Rep 7, 43452 (2017).

66. 66.

Wang, Y., Ma, J. & Xu, Y. et al. The electrical activity of neurons subject to electromagnetic induction and Gaussian White noise. Int J Bifurcat Chaos 27, 1750030 (2017).

67. 67.

Mvogo, A., Takembo, C. N. & Fouda, H. P. E. et al. Pattern formation in diffusive excitable systems under magnetic flow effects. Phys Lett A 381, 2264–2271 (2017).

68. 68.

Yi, G. S., Wang, J. & Deng, B. et al. Morphology controls how hippocampal CA1 pyramidal neuron responds to uniform electric fields: a biophysical modeling study. Sci Rep 7, 3210 (2017).

69. 69.

Ma, J. & Tang, J. A review for dynamics in neuron and neuronal network. Nonlinear Dyn 89, 1569–1578 (2017).

70. 70.

Ma, J. & Tang, J. A review for dynamics of collective behaviors of network of neurons. Sci China Technol Sci 58, 1007–1014 (2015).

71. 71.

Ma, J., Mi, L. & Zhou, P. et al. Phase synchronization between two neurons induced by coupling of electromagnetic field. Appl Math Comput 307, 321–328 (2017).

72. 72.

Wu, F. Q., Wang, C. N. & Jin, W. Y. et al. Dynamical responses in a new neuron model subjected to electromagnetic induction and phase noise. Physica A 469, 81–88 (2017).

73. 73.

Ren, G. D., Zhou, P. & Ma, J. et al. Dynamical response of electrical activities in digital neuron circuit driven by autapse. Int. J. Bifurcat. Chaos 27, 1750187 (2017).

74. 74.

Xu, Y., Jia, Y. & Ma, J. et al. Synchronization between neurons coupled by memristor. Chaos, Solitons & Fractals 104, 435–442 (2017).

75. 75.

Guo, S. L., Xu, Y. & Wang, C. et al. Collective response, synapse coupling and field coupling in neuronal network. Chaos, Solitons & Fractals 105, 120–127 (2017).

## Acknowledgements

This project is supported by National Natural Science Foundation of China under Grants. 11672122, 11365014,11765011. The authors thank Ms. Lulu Lu(Central China Normal University) for her help in plotting Fig. 1.

## Author information

### Affiliations

• Ying Xu
•  & Ya Jia

• Ying Xu
•  & Jun Ma
3. #### Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

• Tasawar Hayat
•  & Ahmed Alsaedi
4. #### Department of Mathematics, Quaid-I-Azam University 45320, Islamabad, 44000, Pakistan

• Tasawar Hayat

### Contributions

Y.X., YJ, J.M. implemented project and analyzed data; T.H. and A.A. extensively shared and discussed results; J.M. wrote the manuscript. All authors read and approved the final manuscript.

### Competing Interests

The authors declare that they have no competing interests.

### Corresponding author

Correspondence to Jun Ma.

### DOI

https://doi.org/10.1038/s41598-018-19858-1

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• , Yuxia Li
•  & Guangyi Wang

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• ### Synchronization realization between two nonlinear circuits via an induction coil coupling

• Zhao Yao
• , Jun Ma
• , Yuangen Yao
•  & Chunni Wang

Nonlinear Dynamics (2019)

• ### Elimination of spiral waves in excitable media by magnetic induction

• Zahra Rostami
• , Sajad Jafari
• , Matjaž Perc
•  & Mitja Slavinec

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# Calculate Stages in Non-Pipelined Processor

I have tried to attempt a question where I have to find the number of stages for non-pipelined processor(8085) for below program :-

MOV #40, R00
MOV #50, R01
MOV #60, R02
SUB R02, R01
INC R01
SUB R00, R02
HLT

What will be the no. of stages and there list here?

• Think about it: How are we supposed to know? It depends on the processor. You know which processor it is, we don't. – gnasher729 Jun 2 '19 at 7:28
• @gnasher729 It is 8085 processor. – nbalodi Jun 2 '19 at 9:00

That processor should come with a manual. If yes, you can use the information in the manual. If no, you can google for the manual, like "8085 manual". The number of cycles and states for each instruction are in chapter 3, for example CPI = 7 states. Then you add up the numbers.

• I am Sorry, but I am not getting your point. I want to know the number of Stages not states, here clocks=44, Instruction count=8, CPI=5.5 and Speed up Factor=.91. I am new to the subject, so please help me in understanding it. – nbalodi Jun 2 '19 at 12:25
• Neither stages nor states is a standard term, but "states" was used in the 1980's. "States" is what the 8085 manual uses. It doesn't use the word "Stage" anywhere. It also doesn't use the word "clocks" but "cycles", so your teacher seems to have changed the terminology. PS. CPI = "ComPare Immediate" instruction. – gnasher729 Jun 2 '19 at 12:38

format: plain


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## Which classes are sets?

From Smullyan and Fitting's Set Theory and the Continuum Problem:

Which classes are sets? Rather than attempt an absolute answer to this (which some authors have done with dubious success), we regard it as philosophically more honest to take these notions as only relative to any given model of the axioms of class-set theory.

This is fine for the text, but it makes me curious what attempts have been made at absolute answers, dubious or not.

-

The attempts to give absolute answers tend to get formalized into theories, and from then on they can be regarded as being relative to the axioms of that theory.

For example, Zermelo-Fraenkel set theory can be based on the "absolute answer" that a class is a set if and only if the ranks (in the cumulative hierarchy) of its members are bounded above, so that there is a level of the cumulative hierarchy at which all those members can be collected into a set.

Similarly, Quine's "New Foundations" uses the idea that a class is a set if (and only if? --- I'm not so sure about this converse) membership in the class can be specified by a stratified formula.

There is a less known but very interesting set theory due to Wilhelm Ackermann (Zur Axiomatik der Mengenlehre. Math. Ann. 131 (1956), 336--345) in which a class is a set if membership in it can be specified by a condition that does not mention the general concept of set (and that uses only sets as parameters).

-
How does Bourbaki's definition of a set using coll_x and the tau operator compare? – Harry Gindi Aug 6 2010 at 15:47
While I appreciate this answer (incidentally, Smullyan and Fitting's book uses NBG) the above quote suggests some think there is an overarching principle that connects all possible definitions. Even if it's wrong as posited, I'm curious what that principle might be. – Jason Dyer Aug 6 2010 at 16:10
I wouldn't say that Quine's approach in NF is "similar", though there's ambiguity in that term. Using a direct restriction on the comprehension principle to dodge the paradox is a fundamentally different approach from the limitation of size approach. Also, I hadn't thought that NF even relies on a set/class distinction in the first place. Does it? – Jeremy Shipley Aug 6 2010 at 16:23
It's true that neither NF nor ZF involves classes, but the original question presupposed them. In both cases, to make sense of the question, one should interpret classes as "virtual" in Quine's sense, i.e., as informal collections $\{x:\phi(x)\}$ of sets satisfying some property $\phi$. And in both cases, the axioms are usually formulated as saying that certain sets exist rather than that certain classes are sets. In both cases, there are extensions incorporating classes explicitly into the theory. Ackermann's theory, in contrast, uses classes from the start. – Andreas Blass Aug 6 2010 at 18:59
@Harry: I won't try to explain Bourbaki's notion of set, partly because I don't remember those axioms precisely enough, and partly because Bourbaki's attitude seems to be far more formalistic than what is presupposed in the question. I believe, though, that Bourbaki's set theory can be viewed as describing a notion of set similar to that of ZF but with (here's the part I don't remember well enough) some restriction on the height of the cumulative hierarchy (perhaps just height $\omega+\omega$). – Andreas Blass Aug 6 2010 at 19:02

It may not completely fit, but - just to have said that - in Zermelo-Fraenkel Set Theory, there are no classes. Classes are a concept of the meta-theory - a "class" is no object inside any model of ZF, it is a collection of sets that can be described somehow, mostly by a proposition with one free variable $A(x)$.

Its a convenient way to write down even the axioms of ZF, but you cannot quantify over them, and every proof of the form "for all classes ..." I know is just by structural induction on formulas.

ZF itself doesnt say anything about classes directly (except maybe that some are not empty). It specifies a few simple given objects and operations you can do on them to create more complicated objects, and these objects are called sets.

Actually, I dont see the point in such discussions. I mean, the reason for not having general comprehension $\{x|A(x)\}$ is that you run into the well-known contradictions (like $\{y|y\not\in y\}\in\{y|y\not\in y\}\rightarrow\{y|y\not\in y\}\not\in\{y|y\not\in y\}$), so one has to restrict the notion of a "set" so far that he still can do anything he needs, but doesnt produce contradictions. And then classes are kept as a concept for collections of sets that cannot be expressed in this notion, when you need them. I dont see the "philosophical" consequences of this.

-


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# The universal affine extension of an abelian variety – Michel Brion (Université Grenoble Alpes)

#### Venue

Dipartimento di Matematica, Aula Magna.

#### Abstract

Every abelian variety $A$ over a field $k$ admits a universal extension by an affine $k$-group scheme. The talk will present a construction of this universal affine extension (first due to Serre when $k$ is algebraically closed of characteristic zero) and discuss its structure, with applications to the category of homogeneous vector bundles over $A$.

Further information is available on the event page on the Indico platform.

Torna in cima


text:
Comment
Share
Q)

# Figure shows the variation of photocurrent with anode potential for a photo-sensitive surface for three different radiations

Let $I_a, I_b\: and \: I_c$ be the intensities and $f_a,f_b\: and \: f_c$ be the frequencies for the curves a,b and c respectively

$\begin {array} {1 1} (a)\;f_a=f_b\: and \: I_a \neq I_b & \quad (b)\;f_a=f_c\: and \: I_a = I_c \\ (c)\;f_a=f_b\: and \: I_a = I_b & \quad (d)\;f_b=f_c\: and \: I_b = I_c \end {array}$

So, $f_a = f_b$
So, $I_a< I_b$


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# Just how to address the formula $x^y=y^x=k$?

For an offered $k>0$ constant, thinking that $x,y>0$. Additionally this formula has a certain name, or some mathematician related to it?

0
2019-05-04 17:39:35
Source Share
WLOG intend that $x \ge y$ and also allow $x = y^n$ for some $n \ge 1$. After that (overlooking $k$) the formula comes to be $y^{ny} = y^{y^n}$, or $ny = y^n$, or $y^{n-1} = n$, which offers
$\displaystyle y = n^{ \frac{1}{n-1} }, x = n^{ \frac{n}{n-1} }$.
Currently it just continues to be to locate the values of $n$ such that $x^y = k$, which you need to simply do numerically. (I presume one have to additionally make up the remedies where $x = y$ ; the chart $x^y = y^x$ has 2 parts which converge at $(e, e)$, which one obtains from the above by taking the restriction as $n \to 1$.)


text:
# Frequently Asked Questions (FAQ)

#### Customer Support 24/7

In addition to our technical support (e.g. via chat), you’ll find resources on our website that may help you with your design using Dlubal Software.

Receive information including news, useful tips, scheduled events, special offers, and vouchers on a regular basis.

• ### How can I display the stresses of a surface via RF‑COM?

New

004484

The stresses of a surface can be displayed via the COM interface. First, you need the interface for the model (IModel) and then the interface for the calculation (ICalculation2). Using this interface, you can get the interface for the results (IResults2):

Sub stresses_surfaces_example()Dim iApp As RFEM5.ApplicationDim iModel As RFEM5.modelSet iModel = GetObject(, "RFEM5.Model")On Error GoTo EIf Not iModel Is Nothing Then    '   get interface from model    Set iApp = iModel.GetApplication    iApp.LockLicense        '   get interface from calculation    Dim iCalc As RFEM5.ICalculation2    Set iCalc = iModel.GetCalculation        '   get interface from results from loadcase 1    Dim iRes As RFEM5.IResults2    Set iRes = iCalc.GetResultsInFeNodes(LoadCaseType, 1)        '   get equivalent stresses    Dim str_equ() As RFEM5.SurfaceEquivalentStresses    str_equ = iRes.GetSurfaceEquivalentStresses(1, AtNo, VonMisesHypothesis)    End IfE:If Err.Number <> 0 Then    MsgBox Err.Number & " " & Err.descriptionEnd IfIf Not iApp Is Nothing Then    iApp.UnlockLicenseEnd If

The GetSurfaceEquivalentStresses function requires the specification of the calculation hypothesis. In this case, the results of the von Mises stress are displayed. Please note that the COM interface uses SI units so the stress is transferred in N/m².

• ### Why do I get the message 28) in RF‑CONCRETE Members, saying that I have to make a "calculation without second order effect" with the internal forces according to the gemetrically linear analysis?

RF-CONCRETE Columns determines the equivalent moment M0e from the moment M02 at the column head and M01 at the column base according to EN 1992‑1‑1, 5.8.8.2 (2), and performs the design according to the model column method with this equivalent moment M0e.

Now, it may happen, for example, that a computationally larger required reinforcement area would result from the cross-section design with the moment M01 at the column head.

To ensure this, message 28) is displayed, according to which the user should perform a standard design with the internal forces according to the linear static analysis. To do this, simply open the RF‑CONCRETE Members add-on module and perform pure design of the internal forces according to the linear static analysis for the member designed in RF‑CONCRETE Columns.

• ### How can I set as quickly as possible that the load combinations generated by the program will be calculated according to the geometrically linear analysis, for example?

004331

In the "Edit Load Cases and Combinations" dialog box, you can specify in the bottom right of the "Combination Expressions" tab which method of analysis should be used as the basis for the generated load combinations. By default, the linear calculation according to the geometrically linear analysis is preset for load cases and the nonlinear calculation according to the second-order analysis for load combinations.

Thus, you can quickly determine whether the load case or the load combination is calculated according to the geometrically linear, second-order, or large deformation analysis. The postcritical analysis option allows you to carry out the stability analysis according to the large deformation analysis with regard to the post-critical failure of the entire structure.

In case the model includes cable members, the calculation according to the large deformation analysis is preset in all cases.
• ### "During the calculation of material non-linearity, the material with a decreasing branch of the diagram can be calculated with one load increment only." Why do I get this error?

Only the default setting of 1 load increment can be set when a complex nonlinear material model is defined. The reason for this is because the program cannot determine the correct material stiffness for each incremental loading amount. The exact maximum load needs to be applied to the structure in order to determine the state of the material's stress/strain diagram.

Figure 01 - Material Model - Nonlinear material defined

This setting can be found and changed under "Calculation Parameters" as well as under the "Calculation Parameters" in the load cases and combinations dialog box.

• ### How can I set the calculation parameters by using the COM interface?

The following code shows how to get different calculation parameters via the COM interface. It also shows how to specify the setting for deactivating shear stiffness:

    '   get model interface    Set iApp = iModel.GetApplication()    iApp.LockLicense        '   get calculation interface    Dim iCalc As RFEM5.ICalculation2    Set iCalc = iModel.GetCalculation        '   get surface bending theory    Dim calc_bend As RFEM5.BendingTheoryType    calc_bend = iCalc.GetBendingTheory        '   get settings for nonlinearities    Dim calc_nl As RFEM5.CalculationNonlinearities    calc_nl = iCalc.GetNonlinearities        '   get precision and tolerance settings    Dim calc_prec As RFEM5.PrecisionAndTolerance    calc_prec = iCalc.GetPrecisionAndTolerance        '   get calculation settings    Dim calc_sets As RFEM5.CalculationSettings    calc_sets = iCalc.GetSettings    'get calculate options    Dim calc_opts As RFEM5.CalculationOptions    calc_opts = iCalc.GetOptions        '   set ShearStiffness to false    calc_opts.ShearStiffness = False    iCalc.SetOptions calc_opts
Under Downloads, you can find the EXCEL macro.
• ### How can I obtain internal forces, calculated in CRANEWAY, for a special x-location?

In CRANEWAY, the internal forces are usually displayed for the max and min locations only. If you want to obtain all internal forces on all x-locations, you have to run a detailed calculation and to disable the "Show only governing results" option. The short video shows how to do it.
• ### What is the difference between the materials Isotropic Plastic 1D and Isotropic Nonlinear Elastic 1D?

The difference between both material models is as follows:

In the Isotropic Nonlinear Elastic 1D material model, no plastic deformations are considered. This means that the material returns to its initial state after the load relief.

In the case of the Isotropic Plastic 1D material model, the plastic deformation is considered.

For both material models, the nonlinear properties are defined in an additional dialog box. When entering data by means of a diagram, it is possible in both models to define the distribution after the last step.

The Isotropic Nonlinear Elastic 1D material model allows for the anti-symmetric input of the stress-strain diagram (different for the positive and negative zone), whereas the isotropic Plastic 1D model only allows for symmetric input.

• ### Which calculation methods are used in RSTAB and RSBUCK?

003564

RSTAB is a FEM program that uses trigonometric trial functions for members. For this reason, it is not necessary to divide the members to obtain sufficiently accurate results and the calculation speed is higher accordingly.

RSBUCK determines the eigenvalues of the stiffness matrix and can thus linearly calculate the critical load and buckling mode of the structure.

• ### Which programs and add-on modules are suitable for the structural analysis and design of steel structures?

Both RFEM and RSTAB provide a suitable solution. For both programs, there are numerous European and international standards as well as various add-on modules that facilitate your daily work in steel construction.

###### Main Programs RFEM or RSTAB
The main programs RFEM or RSTAB are used to define structures, materials, and actions. In addition to the option to create spatial frame and truss structures, such as halls, RFEM also allows for plate, wall, and shell structures, and thus provides a more versatile option. It pays off if it is necessary to also carry out design in other areas, such as solid construction.

###### Available Standards
• EN 1993-1-1 (Eurocode 3),
• AISC according to ANSI/AISC 360 (US standard),
• SIA according to SIA 263:2013 (Swiss standard),
• IS according to IS 800:2007 (Indian standard),
• BS according to BS 5950-1:2000 (British standard) or BS EN 1993-1-1 (British Annex),
• GB according to GB 50017-2003 (Chinese standard),
• CSA according to CSA S16-09 and CSA S16-14 (Canadian standard),
• AS according to AS 4100-1998 + Annex 1 - 1999 (Australian standard),
• NTC-DF according to NTC-RCDF (2004) (Mexican standard),
• SP according to SP 16.13330.2011 (Russian standard),
• SANS according to SANS 10162-1:2011 (South African standard),
• NBR according to ABNT NBR 8800:2008 (Brazilian standard),
• HK according to the standard Code of Practice for the Structural Use of Steel 2011 (Buildings Department - Hong Kong)
• RF-/STEEL - General stress analysis of steel surfaces and members
###### Add-on Modules for Steel Structures

The add-on modules complement the functionality of the main programs. For example, RF‑/STEEL EC3, allows you to perform design of steel structures according to Eurocode 3. The RF‑STEEL Warping Torsion add-on module supplements the design according to Eurocode 3 with torsional buckling analysis with up to 7 degrees of freedom, provided it does not refer to a standard case of EC3.

Further specialized application areas, such as plastic design, a stability analysis according to the eigenvalue method, or the generation of geometric equivalent imperfections and pre-deformed equivalent models are available. Stand-alone modules, such as PLATE-BUCKLING, support your design of rigid or stiffened plates. The SHAPE-THIN program allows you to create any thin-walled cross-sections. Thus, the cross-section properties are determined and stress analyses or plastic designs can be performed.

You can design hinged or rigid connections by using the RF‑/JOINTS add-on modules.

The stand-alone program CRANWAY is available for the design of craneways.

###### Dynamic Analysis

If it is necessary to perform seismic analysis or vibration designs of a building, the RF‑/DYNAM Pro add-on modules provide special tools for determining natural frequencies and mode shapes, for an analysis of forced vibrations, a generation of equivalent loads, or for a nonlinear time history analysis.

If you have any question about the Dlubal Software programs, please do not hesitate to contact our sales department.

• ### When calculating a connection using the FRAME‑JOINT Pro add-on module, a message appears saying that the value is out of the valid range (existing value: 108, minimum value 100, maximum value 100). What does this message mean?

The easiest way is to change to the design notes after the calculation or after the note appears in the design notes (see Figure 02). In that case, this shows that the width of the end plate is not correct. From the input window 1.4.2, it is possible to quickly recognize in the graphic that the value is not within the valid range.

This can be corrected very quickly by adjusting the horizontal bolt spacings (see Figure 03).

1 - 10 of 55

Did you find your question?
If not, contact us via our free e-mail, chat, or forum support, or send us your question via the online form.

#### First Steps

We provide hints and tips to help you get started with the main programs RFEM and RSTAB.

#### Wind Simulation & Wind Load Generation

With the stand-alone program RWIND Simulation, wind flows around simple or complex structures can be simulated by means of a digital wind tunnel.

The generated wind loads acting on these objects can be imported to RFEM or RSTAB.

#### Your support is by far the best

“Thank you for the valuable information.

I would like to pay a compliment to your support team. I am always impressed how quickly and professionally the questions are answered. I have used a lot of software with a support contract in the field of structural analysis, but your support is by far the best. ”


text:
Research

# On q-analogue of a complex summation-integral type operators in compact disks

Ravi P Agarwal1 and Vijay Gupta2*

Author Affiliations

1 Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA

2 School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi-110078, India

For all author emails, please log on.

Journal of Inequalities and Applications 2012, 2012:111 doi:10.1186/1029-242X-2012-111

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/111

 Received: 17 February 2012 Accepted: 22 May 2012 Published: 22 May 2012

© 2012 Agarwal and Gupta; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Recently, Gupta and Wang introduced certain q-Durrmeyer type operators of real variable x ∈ [0, 1] and studied some approximation results in the case of real variables. Here we extend this study to the complex variable for analytic functions in compact disks. We establish the quantitative Voronovskaja type estimate. In this way, we put in evidence the over convergence phenomenon for these q-Durrmeyer polynomials; namely, the extensions of approximation properties (with quantitative estimates) from the real interval [0,1] to compact disks in the complex plane. Some of these results for q = 1 were recently established in Gupta-Yadav.

Mathematical subject classification (2000): 30E10; 41A25.

##### Keywords:
Complex q-Durrmeyer operators; q-integer; q-factorial; q-Beta function; exact order of approximation; quantitative Voronovskaja-type asymptotic formula

### 1. Introduction

In the recent years applications of q-calculus in the area of approximation theory and number theory is an active area of research. Several researchers have proposed the q analogue of exponential, Kantorovich and Durrmeyer type operators. Also Kim [1,2] used q-calculus in area of number theory. Recently, Gupta and Wang [3] proposed certain q-Durrmeyer operators in the case of real variables. The aim of the present article is to extend approximation results for such q-Durrmeyer operators to the complex case. The main contributions for the complex operators are due to Gal; in fact, several important results have been complied in his recent monograph [4]. Also very recently, Gal and Gupta [5-7] have studied some other complex Durrmeyer type operators, which are different from the operators considered in the present article.

We begin with some notations and definitions of q-calculus: For each nonnegative integer k, the q-integer [k]q and the q-factorial [k]q! are defined by

and

respectively. For the integers n, k, n k ≥ 0, the q-binomial coefficients are defined by

In this article, we shall study approximation results for the complex q-Durrmeyer operators (introduced and studied in the case of real variable by Gupta-Wang [3]), defined by

(1.1)

where , n = 1, 2, . . . ; q ∈ (0, 1) and , q-Bernstein basis functions are defined as

also in the above q-Beta functions [8] are given as

Throughout the present article we use the notation and by H(DR), we mean the set of all analytic functions on with for all z DR. The norm ||f||r = max{|f(z)| : |z| r}. We denote πp,n (q; z) = Mn,q (ep; z) for all ep = tp, ∪ {0}.

In what follows, we shall study the approximation properties of the operators Mn,q(f; z), which is extension of the results studied in [10]. Further, for these operators we will estimate an upper bound, a quantitative Voronovskaja-type asymptotic formula, and exact order of approximation on compact disks.

### 2. Basic results

To prove the results of following sections, we need the following basic results.

Lemma 1. Let q ∈ (0, 1). Then, πm,n(q; z) is a polynomial of degree ≤ min (m, n), and

where cs(m) 0 are constants depending on m and q, and Bn,q(f; z) is the q-Bernstein polynomials given by .

Proof. By definition of q-Beta function, with , we have

For m =1, we find

thus the result is true for m = 1 with c1(1) = 1 > 0.

Next for m = 2, with [k + 1]q = 1 + q[k]q, we get

thus the result is true for m = 2 with c1(2) = 1 > 0, c2(2) = q > 0.

Similarly for m = 3, using [k + 2]q = [2]q + q2[k]q and [k + 1]q = 1 + q[k]q we have

where c1(3) = [2]q > 0, c2(3) = 2q2 + q > 0, and c3(3) = q3 > 0.

Continuing in this way the result follows immediately for all m N. □

Lemma 2. Let q ∈ (0, 1). Then, for all m, , we have the inequality

Proof. By Lemma 1, with em = tm, we have

Also

It immediately follows that pn,k(q; 1) = 0, k = 0, 1, 2, . . . , n - 1 and pn,n(q; 1) = 1. Thus, we obtain

Corollary 1. Let r ≥ 1 and q ∈ (0, 1). Then, for all m, and |z| ≤ r we have |πm,n(q; z)| ≤ rm.

Proof. By using the methods Gal [4], p. 61, proof of Theorem 1.5.6, we have |Bn,q (es; z)| ≤ rs, By Lemma 2 and for all and |z| ≤ r,

Lemma 3. Let q ∈ (0, 1) then for , we have the following recurrence relation:

Proof. By simple computation, we have

and

Using these identities, it follows that

Let us denote . Then, the last q-integral becomes

and hence

Therefore,

Finally, using the identity [p + 2]q + [n]q qp+2 = [n + p + 2]q, we get the required recurrence relation. □

### 3. Upper bound

If Pm (z) is a polynomial of degree m, then by the Bernstein inequality and the complex mean value theorem, we have

The following theorem gives the upper bound for the operators (1.1).

Theorem 1. Let for all |z| < R and let 1 ≤ r R, then for all |z| r, q ∈ (0, 1) and ,

where .

Proof First, we shall show that . If we denote with , then by the linearity of Mn,q, we have

Thus, it suffice to show that for any fixed and |z| ≤ r with r ≥ 1, limm→∞ Mn,q(fm, z) = Mn,q(f; z). But this is immediate from limm→∞ ||fm - f||r = 0 and by the inequality

where

Since, π0,n(q; z) = 1, we have

Now using Lemma 3, for all p ≥ 1, we find

However

Combining the above relations and inequalities, we find

From the last inequality, inductively it follows that

Thus, we obtain

which proves the theorem. □

Remark 1. Let q ∈ (0, 1) be fixed. Since, Theorem 1, is not a convergence result. To obtain the convergence one can choose 0 < qn < 1 with qn 1 as n ∞. In that case as n ∞(see Videnskii[9], formula(2.7)), from Theorem 1 we get , uniformly for |z| ≤ r, and for any 1 ≤ r < R.

### 4. Asymptotic formula and exact order

Here we shall present the following quantitative Voronovskaja-type asymptotic result:

Theorem 2. Suppose that f H(DR), R > 1. Then, for any fixed r ∈ [1, R] and for all , |z| ≤ r and q ∈ (0, 1), we have

where , and

Proof. In view of the proof of Theorem 1, we can write . Thus

for all z DR, . If we denote

then Ek,n(q; z) is a polynomial of degree ≤ k, and by simple calculation and using Lemma 3, we have

where

Obviously as 0 < q < 1, it follows that

Next with [n + k + 1]q = [k - 1]q + qk-1[n]q + qn+k-1 + qn+k, we have

and

Thus,

Now we will estimate X3,q,n(k):

Hence, it follows that

Thus,

for all k ≥ 1, and |z| r.

Next, using the estimate in the proof of Theorem 1, we have

for all k, , | z | ≤ r, with 1 ≤ r.

Hence, for all k, , k ≥ 1 and | z | ≤ r, we have

However, since and , it follows that

Now we shall compute an estimate for , k ≥ 1. For this, taking into account the fact that Ek-1,n (q; z) is a polynomial of degree ≤ k - 1, we have

Thus,

and

where

for all | z | ≤ r, k ≥ 1, , where

Hence, for all | z | ≤ r, k ≥ 1,

where Bk,r is a polynomial of degree 3 in k defined as

But E0,n (q; z) = 0, for any z C, and therefore by writing the last inequality for k = 1, 2, . . . we easily obtain step by step the following

Therefore, we can conclude that

As and the series is absolutely convergent in |z| ≤ r, it easily follows that , which implies that This completes the proof of theorem. □

Remark 2. For q ∈ (0, 1) fixed, we have as n ∞, thus Theorem 2 does not provide convergence. But this can be improved by choosing with qn 1 as n ∞. Indeed, since in this case as n → ∞ and from Theorem 2, we get

Our next main result is the exact order of approximation for the operator (1.1).

Theorem 3. Let , , R > 1, and let f H(DR), R > 1. If f is not a polynomial of degree 0, then for any r ∈ [1, R), we have

where the constant Cr(f) > 0 depends on f, r and on the sequence , but it is inde-pendent of n.

Proof. For all and , we have

We use the following property.

to obtain

By the hypothesis, f is not a polynomial of degree 0 in DR, we get ||e1(1-e1)f"-2e1 f'||r > 0. Supposing the contrary, it follows that z(1 - z)f"(z) - 2zf'(z) = 0 for all |z| r, that is (1 - z)f"(z) - 2f'(z) = 0 for all | z| r with z ≠ 0. The last equality is equivalent to [(1 - z) f'(z)]' - f'(z) = 0, for all |z| r with z ≠ 0. Therefore, (1 - z) f'(z) - f(z) = C, where C is a constant, that is, , for all |z| r with z ≠ 0. But since f is analytic in and r ≥ 1, we necessarily have C = 0, a contradiction to the hypothesis.

But by Remark 2, we have

with as n . Therefore, it follows that there exists an index n0 depending only on f, r and on the sequence (qn)n, such that for all n n0, we have

which implies that

For 1 ≤ n n0 - 1, we clearly have

where , which finally implies

where

Combining Theorem 3 with Theorem 1 we get the following.

Corollary 2. Let for all , R > 1, and suppose that f H(DR). If f is not a polynomial of degree 0, then for any r ∈ [1, R), we have

where the constants in the above equivalence depend on f, r, (qn)n, but are independent of n.

The proof follows along the lines of [7].

Remark 3. For 0 ≤ α ≤ β, we can define the Stancu type generalization of the operators (1.1) as

The analogous results can be obtained for such operators. As analysis is different, it may be considered elsewhere.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors read and approved the final manuscript.

### Acknowledgements

The authors are thankful to the referees for valuable suggestions leading to overall improvements in article.

### References

1. Kim, T: Some Identities on the q-integral representation of the product of several q-Bernstein-type polynomials, abstract and applied analysis. Article ID 634675, 11 (2011). doi:10.1155/2011/634675 (2011)

2. Kim, T: q-generalized Euler numbers and polynomials. Russ J Math Phys. 13(3), 293–298 (2006). Publisher Full Text

3. Gupta, V, Wang, H: The rate of convergence of q-Durrmeyer operators for 0 < q <1. Math Meth Appl Sci. 31, 1946–1955 (2008). Publisher Full Text

4. Gal, SG: Approximation by Complex Bernstein and Convolution-Type Operators. World Scientific Publ Co, Singapore (2009)

5. Gal, SG, Gupta, V: Approximation by a Durrmeyer-type operator in compact disk. Annali Dell Universita di Ferrara. 57, 261–274 (2011). Publisher Full Text

6. Gal, SG, Gupta, V: Quantative estimates for a new complex Durrmeyer operator in compact disks. Appl Math Comput. 218(6), 2944–2951 (2011)

7. Gal, S, Gupta, V, Mahmudov, NI: Approximation by a complex q Durrmeyer type operators. Ann Univ Ferrara. 58(1), 65–82 (2012). Publisher Full Text

8. Kac, V, Cheung, P: Quantum Calculus, Springer, New York (2002)

9. Videnskii, VS: On q-Bernstein polynomials and related positive linear operators (in Russian). Problems of Modern Mathematics and Mathematical Education, St.-Petersburg. 118–126 (2004)

10. Gupta, V, Yadav, R: Approximation by a complex summation-integral type operator in compact disks. In: Mathematica Slovaca (to appear)


text:
# Find the eigenvalues of a projection operator

A projection operator $P$ is defined as $P^2$=$P$. Use this definition to find the eigenvalues of this operator.

In this question is it necessary to define what the projection operator is? And won't the eigenvalue just be zero?

Let $\lambda$ be an eigenvalue of $P$ for the eigenvector $v$. You have $\lambda^2 v = P^2 v = P v = \lambda v$. Because $v \neq 0$ it must be $\lambda^2 = \lambda$. The solutions of the last equation are $\lambda_1 = 0$ and $\lambda_2 = 1$. Those are the only possible eigenvalues the projection might have...
If $\mathbf x$ is an eigenvector of $P$ with eigenvalue $\lambda$, then $P(\mathbf x) = \lambda \mathbf x$. If $P^2 = P(P) = P$, then $P(P(\mathbf x)) = P(\lambda \mathbf x) = \lambda^2 \mathbf x$, but this must also equal $\lambda \mathbf x$. Therefore the eigenvalues of $P$ can only be members of your base field such that $\lambda^2 = \lambda$.
Assuming your base field is $\Bbb R$, there's one more value that satisfies this equation besides $\lambda =0$. What is it?
The eigenvalues are $0$ and $1$. Indeed, we know that for a projector $P$ defined on a vector space $E$, we have $$E= \ker P\oplus \operatorname{im}P=\ker P\oplus \ker(I-P)$$ $\ker P$ is the eigenspace associated with the eigenvalue $0$, $\ker(I- P)$ the eigenspace associated with the eigenvalue $1$. As $E$ is the direct sum of these eigenspaces, we have all eigenspaces, and all eigenvalues.


text:
## anonymous 3 years ago HELP PLEASE! FAN AND MEDAL FOR BEST HELPER!!!!

1. anonymous

what question

2. anonymous

3. anonymous

try all fractions where the numerator divides 9 and the denominator divides 2

4. anonymous

they are $\pm1,\pm3,\pm9,\pm\frac{1}{2},\pm\frac{3}{2},\pm\frac{9}{2}$

5. anonymous

thanks for the help i got it now... :D


text:
3 deleted 2 characters in body

There seems to be general opinion that, for positive integral quadratic forms in at least three variables, spinor genera in the same genus all have the same mass (not representation measures of some number, that is different, indeed some recent authors write of representation mass of numbers and it throws me off). Authors R. Scharlau and R. Schulze-Pillot have been loosely mentioned in this regard.

Does anyone know for sure, and in particular know a simple (published) reference for positive forms? Evidently there is an analogous formulation where it is true for indefinite forms, see The Hirzebruch-Mumford volume for the orthogonal group and applications, by Valery Gritsenko, Klaus Hulek, G.K. Sankaran,

http://arxiv.org/abs/math/0512595

I should add that Kneser (1961) is sometimes mentioned in this regard, but my take is that, while his methods can be used to reconstruct the result, he is not explicit about the mass.

Here is computer output on an example that appears in Benham, Earnest, Hsia, Hung (1990), formula (3.8), positive ternary forms, including one of the 29 known spinor regular forms that are not regular. The sextuple a b c d e f refers to the form $T(x,y,z) = a x^2 + b y^2 + c z^2 + d y z + e z x + f x y,$ with discriminant $\Delta = 4 a b c + d e f - a d^2 - b e^2 - c f^2.$

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

===Discriminant   343 ==Genus Size==  3
343 = 7^3
All forms in regular spinor genus represent      1
-----------------------------
Spinor genus misses        1     4    16    64   121   256
484   529   841
343:    2     7          8      7    1    0    auto 4   Level 196  irreg spin candidate
--------------------------size 1
Spinor genus misses     no exceptions
343:    1     2         49      0    0    1    auto 8   Level 196
343:    1     7         14      7    0    0    auto 8   Level 196
--------------------------size 2
Disc    343
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

There seems to be general opinion that, for positive integral quadratic forms in at least three variables, spinor genera in the same genus all have the same mass (not representation measures of some number, that is different, indeed some recent authors write of representation mass of numbers and it throws me off). Authors R. Scharlau and R. Schulze-Pillot have been loosely mentioned in this regard.

Does anyone know for sure, and in particular know a simple (published) reference for positive forms? Evidently there is an analogous formulation where it is true for indefinite forms, see The Hirzebruch-Mumford volume for the orthogonal group and applications, by Valery Gritsenko, Klaus Hulek, G.K. Sankaran,

http://arxiv.org/abs/math/0512595

I should add that Kneser (1961) is sometimes mentioned in this regard, but my take is that, while his methods can be used to reconstruct the result, he is not explicit about the mass.

Here is computer output on an example that appears in Benham, Earnest, Hsia, Hung (1990), formula (3.8), positive ternary forms, including one of the 29 known spinor regular forms that are not regular. The sextuple refers to the form $T(x,y,z) = a x^2 + b y^2 + c z^2 + d y z + e z x + f x y,$ with discriminant $\Delta = 4 a b c + d e f - a d^2 - b e^2 - c f^2.$

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

===Discriminant   343 ==Genus Size==  3
343 = 7^3
All forms in regular spinor genus represent      1
-----------------------------
Spinor genus misses        1     4    16    64   121   256
484   529   841
343:    2     7          8      7    1    0    auto 4   Level 196  irreg spin candidate
--------------------------size 1
Spinor genus misses     no exceptions
343:    1     2         49      0    0    1    auto 8   Level 196
343:    1     7         14      7    0    0    auto 8   Level 196
--------------------------size 2
Disc    343
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

1

# Mass of spinor genus, positive integral quadratic forms

There seems to be general opinion that, for positive integral quadratic forms in at least three variables, spinor genera in the same genus all have the same mass (not representation measures of some number, that is different, indeed some recent authors write of representation mass of numbers and it throws me off). Authors R. Scharlau and R. Schulze-Pillot have been loosely mentioned in this regard.

Does anyone know for sure, and in particular know a simple (published) reference for positive forms? Evidently there is an analogous formulation where it is true for indefinite forms, see The Hirzebruch-Mumford volume for the orthogonal group and applications, by Valery Gritsenko, Klaus Hulek, G.K. Sankaran,

http://arxiv.org/abs/math/0512595

I should add that Kneser (1961) is sometimes mentioned in this regard, but my take is that, while his methods can be used to reconstruct the result, he is not explicit about the mass.


text:
# Simple circuit with transistor using Raspberry Pi

I'm trying to use a Raspberry Pi to control a set of 4 LEDs. I'd like to control the LEDs in software by using a GPIO pin. Here is the schematic I've come up with... (please ignore the wire in the top right corner of the diagram, the transistor should make or break the circuit)

I'll be using a 9V battery to power 4 LEDs in series. Each LED has a voltage drop of 2V and a current rating of 20mA. Therefore my resistor should have the value of 50 Ohms (1V / 20 x 10^-3A). The diagram has an incorrect resistor value shown.

My problem is that I'm not sure of what will happen when I connect the Pi's pin to the base of the transistor. Will it mean that the pin becomes connected to the 9V circuit and will draw 20mA after I set the pin to high? Or does the pins voltage have its own influence? (the pin operates at 3.3V).

There is probably a better way to do this but I'm trying to learn about transistors so I don't want to remove the transistor.

• Do you understand that the collector and the emitter are shorted on the schematic? – Eugene Sh. Aug 9 '16 at 19:16
• @EugeneSh. Yes, I noted in my question that the wire in the top right corner is erroneous and should be ignored – CS Student Aug 9 '16 at 19:29
• And the reason you're not spending the same amount of energy on properly drawing your intentions as you expect people here to put into answering, is what? – Asmyldof Aug 9 '16 at 22:28

This is how you should connect the NPN transistor:

R_GPIO must limit base current to about ten times LED current divided by transistor gain. Assuming a gain of 150 at the specified collector current:

$$R_{GPIO} = {{3.3 V - 0.7 V} \over {10 \cdot {20 mA \over 150}}} \approx 2 k$$

• Does the formula you have used have a name? I've never seen it before and would like to know more about it – CS Student Aug 9 '16 at 21:50
• The formula is just a slightly disguised form of Ohm's Law: R = V/I. The voltage across the base resistor is the GPIO voltage minus the transistor's base-emitter voltage. The required base current to put the transistor in saturation is 10 times [LED current over the transistor's gain]. – Peter Bennett Aug 10 '16 at 5:33

A correct way to do this is:

simulate this circuit – Schematic created using CircuitLab

In an NPN transistor, the voltage between emitter and base will never be more than about 0.7 volts. In your circuit, if a High from the Pi is 3.3 volts, that would leave the emitter at 2.6 volts - not enough to turn on the LEDs.

My way, the collector/emitter voltage should be about 0.3 volts when the Pi output is High. The base resistor limits the current drawn from the Pi output when it is High.

• Second-to-last paragraph doesn't follow logically - the voltage can't be more than 0.7 volts, but you didn't say it can't be less - if it was -5.7V or less then that would be enough for the LEDs to light. – immibis Aug 9 '16 at 21:01
• Unfortunately I'm quite confused by your answer. When you say the collector/emitter should be at around 0.3V have you calculated that by subtracting the voltages across the LEDs and resistor from the batteries 9 volts? Also why is the transistor connected to both grounds? I'm unsure as to what happens at the point the emitter pin contacts the battery and Pi ground wire (what happens to the voltage / current?) – CS Student Aug 9 '16 at 21:48
• @CSStudent -- minimum Vce gets approached as collection current grows. It is a property of transistors, and the number is listed in the data sheet. Generally, though, it is around 0.3. All you need to do is drive the transistor hard enough to get it there – Scott Seidman Aug 9 '16 at 22:21
• @immibis In order to get a -5.7V, you would need to connect the +9V terminal to the +3.3V net of the RPi, and you could certainly light up the LEDs that way.... but you wouldn't be able to turn them off. – W5VO Aug 10 '16 at 4:12
• @W5VO I didn't say it wasn't true; I said it didn't follow logically. (And actually, if you did that, you would be able to turn them mostly off by applying logic low = 5.7V to the base) – immibis Aug 10 '16 at 4:57

This circuit would not work, even if the collector and emitter were not shorted. Assuming around a 2V voltage drop across each diode when they're on, the emitter would thus be sitting at about 8V -- even if the resistor were shorted out. The base needs to be about 0.7V higher than then the emitter for the transistor to stay on -- about 8.7V. The Pi has 3.3V outputs, and thus isn't big enough to keep the transistor turned on

• Ah I wasn't aware of the 0.7V difference needed, I just thought some kind of current was needed to trigger the transistor but didn't know any exact values. Is there any way for me to somehow get the 3.3V output and step it up to the 8.7V trigger needed? – CS Student Aug 9 '16 at 19:31
• @CSStudent Yes, Peter Bennet posted the right way, which is to put the led's on the high side. – Scott Seidman Aug 9 '16 at 20:27


text:
My proof that sum of convergent sequences converges to sum of limits

Does my proof appear correct? Also, do you like the notation?

$\textbf{Theorem.}$ If $(a_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in \mathbb{N}}$ are convergent real sequences, then $$\lim_{n \to \infty} \left( a_n + b_n \right) = \left( \lim_{n \to \infty} a_n \right) + \left( \lim_{n \to \infty} b_n \right) .$$ $\textit{Proof.}$ Let $\varepsilon > 0$. It remains to prove that there is $N \in \mathbb{N}$ such that, for every natural number $m > N$, we have $$\left| a_m + b_m - \left( \lim_{n \to \infty} a_n \right) - \left( \lim_{n \to \infty} b_n \right) \right| < \varepsilon .$$ By assumption, there is $N_a \in \mathbb{N}$ such that, for every natural number $m > N_a$, we have $$\left| a_m - \left( \lim_{n \to \infty} a_n \right) \right| < \varepsilon / 2 .$$ Analogously, there is $N_b \in \mathbb{N}$ such that, for every natural number $m > N_b$, we have $$\left| b_m - \left( \lim_{n \to \infty} b_n \right) \right| < \varepsilon / 2 .$$ We choose $N := \max(N_a, N_b)$. Let $m \in \mathbb{N}$ such that $m > N$. Obviously, for this $m$, each of the above two inequalities holds. Thus, we may add the inequalities. Doing so, we obtain \begin{equation*} \begin{split} \varepsilon = \varepsilon / 2 + \varepsilon / 2 & > \left| a_m - \left( \lim_{n \to \infty} a_n \right) \right| + \left| b_m - \left( \lim_{n \to \infty} b_n \right) \right| \\ & \ge \left| a_m - \left( \lim_{n \to \infty} a_n \right) + b_m - \left( \lim_{n \to \infty} b_n \right) \right| &\quad& \text{by subadditivity of abs. val.} \\ & = \left| a_m + b_m - \left( \lim_{n \to \infty} a_n \right) - \left( \lim_{n \to \infty} b_n \right) \right| . \end{split} \end{equation*} Hence, by transitivity, \begin{equation*} \varepsilon > \left| a_m + b_m - \left( \lim_{n \to \infty} a_n \right) - \left( \lim_{n \to \infty} b_n \right) \right|, \end{equation*} QED.

• Your proof is correct. I think that notation would be lighter if you call your limits $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$, however this is just a matter of taste. – Crostul Mar 31 '15 at 11:22
• I assume the proof doesn't require any differences to make it apply to pointwise or uniform convergent sequences. – mavavilj Sep 30 '16 at 16:40

Your proof is correct. ${ }$


text:
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## G = C32×D8order 144 = 24·32

### Direct product of C32 and D8

direct product, metacyclic, nilpotent (class 3), monomial

Aliases: C32×D8, C243C6, C4.1C62, D4⋊(C3×C6), C81(C3×C6), (C3×C24)⋊5C2, (C3×D4)⋊4C6, (C3×C6).41D4, C6.20(C3×D4), C12.23(C2×C6), (D4×C32)⋊7C2, C2.3(D4×C32), (C3×C12).50C22, SmallGroup(144,106)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C32×D8
 Chief series C1 — C2 — C4 — C12 — C3×C12 — D4×C32 — C32×D8
 Lower central C1 — C2 — C4 — C32×D8
 Upper central C1 — C3×C6 — C3×C12 — C32×D8

Generators and relations for C32×D8
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 114 in 66 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2 [×2], C3 [×4], C4, C22 [×2], C6 [×4], C6 [×8], C8, D4 [×2], C32, C12 [×4], C2×C6 [×8], D8, C3×C6, C3×C6 [×2], C24 [×4], C3×D4 [×8], C3×C12, C62 [×2], C3×D8 [×4], C3×C24, D4×C32 [×2], C32×D8
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], D4, C32, C2×C6 [×4], D8, C3×C6 [×3], C3×D4 [×4], C62, C3×D8 [×4], D4×C32, C32×D8

Smallest permutation representation of C32×D8
On 72 points
Generators in S72
(1 34 61)(2 35 62)(3 36 63)(4 37 64)(5 38 57)(6 39 58)(7 40 59)(8 33 60)(9 69 42)(10 70 43)(11 71 44)(12 72 45)(13 65 46)(14 66 47)(15 67 48)(16 68 41)(17 31 51)(18 32 52)(19 25 53)(20 26 54)(21 27 55)(22 28 56)(23 29 49)(24 30 50)
(1 31 9)(2 32 10)(3 25 11)(4 26 12)(5 27 13)(6 28 14)(7 29 15)(8 30 16)(17 42 61)(18 43 62)(19 44 63)(20 45 64)(21 46 57)(22 47 58)(23 48 59)(24 41 60)(33 50 68)(34 51 69)(35 52 70)(36 53 71)(37 54 72)(38 55 65)(39 56 66)(40 49 67)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(18 24)(19 23)(20 22)(25 29)(26 28)(30 32)(33 35)(36 40)(37 39)(41 43)(44 48)(45 47)(49 53)(50 52)(54 56)(58 64)(59 63)(60 62)(66 72)(67 71)(68 70)

G:=sub<Sym(72)| (1,34,61)(2,35,62)(3,36,63)(4,37,64)(5,38,57)(6,39,58)(7,40,59)(8,33,60)(9,69,42)(10,70,43)(11,71,44)(12,72,45)(13,65,46)(14,66,47)(15,67,48)(16,68,41)(17,31,51)(18,32,52)(19,25,53)(20,26,54)(21,27,55)(22,28,56)(23,29,49)(24,30,50), (1,31,9)(2,32,10)(3,25,11)(4,26,12)(5,27,13)(6,28,14)(7,29,15)(8,30,16)(17,42,61)(18,43,62)(19,44,63)(20,45,64)(21,46,57)(22,47,58)(23,48,59)(24,41,60)(33,50,68)(34,51,69)(35,52,70)(36,53,71)(37,54,72)(38,55,65)(39,56,66)(40,49,67), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,8)(3,7)(4,6)(10,16)(11,15)(12,14)(18,24)(19,23)(20,22)(25,29)(26,28)(30,32)(33,35)(36,40)(37,39)(41,43)(44,48)(45,47)(49,53)(50,52)(54,56)(58,64)(59,63)(60,62)(66,72)(67,71)(68,70)>;

G:=Group( (1,34,61)(2,35,62)(3,36,63)(4,37,64)(5,38,57)(6,39,58)(7,40,59)(8,33,60)(9,69,42)(10,70,43)(11,71,44)(12,72,45)(13,65,46)(14,66,47)(15,67,48)(16,68,41)(17,31,51)(18,32,52)(19,25,53)(20,26,54)(21,27,55)(22,28,56)(23,29,49)(24,30,50), (1,31,9)(2,32,10)(3,25,11)(4,26,12)(5,27,13)(6,28,14)(7,29,15)(8,30,16)(17,42,61)(18,43,62)(19,44,63)(20,45,64)(21,46,57)(22,47,58)(23,48,59)(24,41,60)(33,50,68)(34,51,69)(35,52,70)(36,53,71)(37,54,72)(38,55,65)(39,56,66)(40,49,67), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,8)(3,7)(4,6)(10,16)(11,15)(12,14)(18,24)(19,23)(20,22)(25,29)(26,28)(30,32)(33,35)(36,40)(37,39)(41,43)(44,48)(45,47)(49,53)(50,52)(54,56)(58,64)(59,63)(60,62)(66,72)(67,71)(68,70) );

G=PermutationGroup([(1,34,61),(2,35,62),(3,36,63),(4,37,64),(5,38,57),(6,39,58),(7,40,59),(8,33,60),(9,69,42),(10,70,43),(11,71,44),(12,72,45),(13,65,46),(14,66,47),(15,67,48),(16,68,41),(17,31,51),(18,32,52),(19,25,53),(20,26,54),(21,27,55),(22,28,56),(23,29,49),(24,30,50)], [(1,31,9),(2,32,10),(3,25,11),(4,26,12),(5,27,13),(6,28,14),(7,29,15),(8,30,16),(17,42,61),(18,43,62),(19,44,63),(20,45,64),(21,46,57),(22,47,58),(23,48,59),(24,41,60),(33,50,68),(34,51,69),(35,52,70),(36,53,71),(37,54,72),(38,55,65),(39,56,66),(40,49,67)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72)], [(2,8),(3,7),(4,6),(10,16),(11,15),(12,14),(18,24),(19,23),(20,22),(25,29),(26,28),(30,32),(33,35),(36,40),(37,39),(41,43),(44,48),(45,47),(49,53),(50,52),(54,56),(58,64),(59,63),(60,62),(66,72),(67,71),(68,70)])

C32×D8 is a maximal subgroup of   C327D16  C328SD32  C248D6  C24.26D6

63 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4 6A ··· 6H 6I ··· 6X 8A 8B 12A ··· 12H 24A ··· 24P order 1 2 2 2 3 ··· 3 4 6 ··· 6 6 ··· 6 8 8 12 ··· 12 24 ··· 24 size 1 1 4 4 1 ··· 1 2 1 ··· 1 4 ··· 4 2 2 2 ··· 2 2 ··· 2

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D4 D8 C3×D4 C3×D8 kernel C32×D8 C3×C24 D4×C32 C3×D8 C24 C3×D4 C3×C6 C32 C6 C3 # reps 1 1 2 8 8 16 1 2 8 16

Matrix representation of C32×D8 in GL3(𝔽73) generated by

 8 0 0 0 8 0 0 0 8
,
 1 0 0 0 64 0 0 0 64
,
 72 0 0 0 16 57 0 16 16
,
 72 0 0 0 1 0 0 0 72
G:=sub<GL(3,GF(73))| [8,0,0,0,8,0,0,0,8],[1,0,0,0,64,0,0,0,64],[72,0,0,0,16,16,0,57,16],[72,0,0,0,1,0,0,0,72] >;

C32×D8 in GAP, Magma, Sage, TeX

C_3^2\times D_8
% in TeX

G:=Group("C3^2xD8");
// GroupNames label

G:=SmallGroup(144,106);
// by ID

G=gap.SmallGroup(144,106);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-2,457,3244,1630,88]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽


text:
Are subatomic movements random

1. Jun 3, 2013

susskind99

In the Cosmic Landscape, Susskind writes:

My question is if subatomic movements are random then does the conservation of momentum law break down at the quantum level? My hunch is yes. The conservation laws only apply at the classical level. Some people say that Noether's Theorem proves the conservation laws but that was devised in 1918 before the HUP was devised.

2. Jun 3, 2013

Whenever a particle is confined it doesn't have a well defined momentum. Momentum in a certain understanding doesn't make sense for a confined particle. The Schrödinger equation conserves momentum therefore it is conserved, but if you start with a particle which is confined the equation will conserve the probability distribution of the momentum. The particle doesn't have one momentum which changes but it never had a well defined momentum to begin with.

3. Jun 3, 2013

VantagePoint72

Classical conservation laws still apply at the quantum level.

If you prepare a state in a momentum eigenstate and the momentum operator commutes with the Hamiltonian (the requirement for momentum conservation), then it will stay in that eigenstate as it time-evolves. Since, in practice, we don't work with particles in momentum eigenstates, there is uncertainty about its value before a measurement is made. Thus, we usually discuss conservation laws in terms of the expectation values of the corresponding operators.

In the Heisenberg picture of quantum mechanics (where operators depend on time), the momentum operator associated with the system is exactly conserved when the corresponding classical momentum would be. The proof comes via Noether's theorem, just as in the classical case. As a consequence, the expectation value of momentum measurements also obey conservation of momentum, and it evolves in time due to external forces exactly as described by Newton's second law law. Similarly for quantum fields, with the appropriate generalization to relativistic momentum. Every particle is associated with a quantum field, and that field has a conjugate momentum that is conserved in the same circumstances as a classical field would be. Noether's theorem is the proof.

4. Jun 3, 2013

daveyrocket

Noether's theorem derives conservation laws from symmetries of the Hamiltonian (or Lagrangian). If you take the traditional example of dropping a ball with a constant gravitational field in the y direction, $H = p_x^2/2m + p_y^2/2m + mgy$, the you will find with Noether's theorem that momentum in the y direction isn't conserved because of the external potential. The key idea here is if you neglect the change in momentum of the earth that occurs when the ball's gravity pulls on the earth, you no longer get conservation of momentum (in that direction).

If you rewrite the example to be contain the earth and use a central potential between the earth and the ball, $H = p_b^2/2m_b + p_E^2/2m_E + Gm_bm_E/(r_b - r_E)$ then you find that total momentum is conserved via Noether's theorem. In classical mechanics you check the Poisson bracket {P,H} where P = p_b + p_E.

The same thing is true in quantum mechanics. If you have an external potential, you won't get any momentum eigenstate so you won't have momentum be conserved. This is because your particle is constantly interacting with its environment through the potential or the boundary conditions and you've neglected any effect the particle has on the environment. Thus, momentum is not conserved.

If you write a quantum Hamiltonian with two particles that interact with each other through a central potential, you will find that the total momentum is conserved. This is because $[P,H] = [P,V(r_1-r_2)] = [p_1,V(r_1-r_2)] + [p_2,V(r_1-r_2)] = 0$. That last quantity is zero because the potential depends only on the difference $r_1-r_2$ which, if you assume any sort of series expansion of V you will find that $[p_1,V(r_1-r_2)] = -[p_2,V(r_1-r_2)]$ since $[p_1,r_2] = 0$.

5. Jun 3, 2013

Chronos

The problem is nearly all of physics involves large numbers of particles. Statistical mechanics is the only known comprehensible way to model such systems. The motion of individual particles may be deterministic, but, rapidly exceeds human ability to comprehend when other particles are added to the mix. It is still frightfully difficult to analytically solve even 'simple' 3 body problems.

6. Jun 4, 2013

susskind99

I have a real hard time believing that. If the momentum cannot even be defined at the quantum level then how do you know it's deterministic? Plus in the text Susskind clearly says that atomic movements are random.

Also, to other members, I didn't get an answer regarding whether subatomic movements are random.

7. Jun 4, 2013

daveyrocket

The thing is, when you say "subatomic movements are random" it implies that things like position and momentum have well defined values all the time and you can track them well enough to say "see, the motion is random!" But that isn't true. You can't identify these quantities without taking a measurement, and taking a measurement perturbs the system (causes a wavefunction collapse). The results of measurements are random, although the statistics of those measurements is determined by the wavefunction.

8. Jun 4, 2013

Staff: Mentor

Yea - interesting questions.

First what subatomic objects are doing, or even what properties they have, when not being observed QM is silent about - but various interpretations have different takes.

Also when you observe it in general you cant predict the outcome - only probabilities. These are determined by the system state and a mathematical object called an operator that depends on what you are measuring. The operator for the total momentum (or you can consider the operator fixed and the state changing) is in general conserved - ie remains the same. But the actual measurements are still only given by probabilities.

Thanks
Bill

Last edited: Jun 4, 2013
9. Jun 4, 2013

susskind99

See, that's what I thought. If you can't know the properties of an object then how do they know that their movements are random?

At the same time, the structure that gives rise to the galaxies supposedly is determined by the UP. And the distribution of the galaxies throughout the universe is random so it stands to reason that that which galaxies are composed of is also random.

10. Jun 4, 2013

Crazymechanic

Maybe it is not random and there is a great chance it isn't , but to us and our atleast current understanding it seems random , just like when you were a child many things seemed different to you than they seem now when you grow up.
Some things never make sense , that's life.

11. Jun 4, 2013

VantagePoint72

You didn't get an answer to that because it's not what you asked. What you asked was:

That is a very different question, and is completely independent of whether or not quantum motion is truly random. The answer is that even if they are truly random (in sense permitted by QM's formalism) then conservation of momentum still doesn't break down at the quantum level.

12. Jun 5, 2013

daveyrocket

Saying that atomic movements are random is illustrative, but imprecise. In the paragraphs that you quote in the original post, the author is making an analogy with classical physics.

In classical physics, you can construct a model of atomic motions by taking a bunch of billiard balls and giving them some sort of attractive potential (pairwise is easiest). If you do this, you will see that at temperature their motions are more-or-less random. They're not actually random, they're governed by Newton's laws, but with many particles, the correlations in their movements have a very short time scale. So their motions might as well be random, and as far as doing statistical mechanics is concerned, that's a reasonably good approximation for the right temperatures and pressures.

If one considers a harmonic approximation for the interatomic potentials, then in the classical system you can compute the normal modes of the system of atoms. These are essentially the standing waves of the atoms. These are also the quantum vibrational modes of the system of atoms. So the analogy with classical physics is not so farfetched. The only thing is, the quantum oscillator does not run out of kinetic energy at zero temperature.

13. Jun 7, 2013

Khashishi

Conservation of momentum is not broken by uncertainty. That's a misunderstanding of how uncertainty works. If we have a closed box with two particles inside, and we know for certain that the total momentum in the box is 2, if we have uncertainty in the momentum of one particle, we have a corresponding uncertainty in the other particle. If we know one particle has a 20% amplitude of having a momentum of 1.5, then the other particle has a 20% amplitude of having a momentum of 0.5, but the total is still 2. If we have uncertainty in the total momentum of the box, that uncertainty is still conserved if you consider a bigger box where the momentum is known. The momenta of the contents of the box will still add up in a mathematically consistent way. (Think of the wavefunction as a set of possible scenarios. Each possible scenario conserves momentum, but you don't know which scenario you are in, or you are in some combination of multiple scenarios.)