Tian Ma
email: matian56@sina.com

Shouhong Wang
email: showang@indiana.edu

TOPOLOGICAL PHASE TRANSITION III: SOLAR SURFACE ERUPTIONS AND SUNSPOTS

Keywords: topological phase transition, sunspot, solar flare, solar prominence, solar eruption, temperature blow-up, statistical theory of heat, energy level formula of temperature, photon number entropy formula, anti-diffusive effect of heat, Fourier law, Fick law, heat transfer, photon absorption, photon radiation

. The anti-diffusive effect of heat states that due to the higher rate of photon absorption and emission of the particles with higher energy levels, the photon flux will move toward to the higher temperature regions from the lower temperature regions. This antidiffusive effect of heat leads to a modified law of heat transfer, which includes a reversed heat flux counteracting the heat diffusion. It is this anti-diffusive effect of heat and thereby the modified law of heat transfer that lead to the temperature blow-up and consequently the formation of sunspots, solar eruptions, and solar prominences. This anti-diffusive effect of heat may be utilized to design a plasma instrument, directly converting solar energy into thermal energy. This may likely offer a new form of fuel much more efficient than the photovoltaic devices.

The main objective of this paper is to provide a new theory for the formation of the solar surface eruptions and sunspots. This is part of the research program initiated recently by the authors on theory and applications of topological phase transitions, including (1) quantum phase transitions [START_REF]Topological Phase Transitions I: Quantum Phase Transitions[END_REF],

(2) formation of galactic spiral structure [START_REF]Topological Phase Transitions II: Spiral Structure of Galaxies[END_REF],

(3) boundary-layer separation of fluid flows, and (4) interior separation of fluid flows.

The Sun is mainly made up of hydrogen and helium, and is entirely in a gaseous state. Solar structure is divided into two parts: the interior and the atmosphere. The atmosphere is composed of three distinct parts: the photosphere, the chromosphere and the corona, and the most visible light comes from the photosphere. Astronomical observations reveal that the solar flare, the solar prominence and the sunspots are important phenomena for the solar atmosphere. Solar flares refer to sudden flashes in Sun's surface brightness. A solar prominence is a large, bright gaseous feature extending outward from the Sun's surface, usually in a loop shape. Sunspots are the areas on Sun's surface, which are darker than their surrounding areas. Astronomical observations show that solar flares, prominences and sunspots are intimately related. The theory established in this paper verifies this claim, and explains their relations.

The main ingredients of the paper are as follows.

First, the most important ingredient of the study present in this paper is the law of heat transfer based on the recently developed statistical theory of heat by the authors [START_REF] Ma | Statistical theory of heat[END_REF]. In this theory, we derived the energy level temperature formula, showing that the temperature is essentially the average energy level of system particles. We also obtained the photon number entropy formula, demonstrating that the entropy is the number of photons in the gap between system particles, and the physical carrier of heat is the photons.

Another important component of the theory is the vibratory mechanism of photon absorption and radiation: a particle can only absorb and radiate photons while experiencing vibratory motion. The higher the frequency of the vibration of the particle, the larger the absorbing and radiating energy. The vibration or irregular motion of particles in a system is caused by collisions between particles and by absorbing and radiating photons.

This mechanism shows immediately that for particles in high speed vibration and irregular motion, the rate of photon emission and absorption increases, leading to the number density of photons to increase, and further causing the particle energy levels to elevate. Hence, the photon absorption and emission induce the concentration of temperature, which we call the anti-diffusive effect of heat:

(1.1)
Due to the higher rate of photon absorption and emission of the particles with higher energy levels, the photon flux will move toward the higher temperature regions from the lower temperature regions.

By the Stefan-Boltzmann law, the reversed heat flux measuring the anti-diffusive effect is expressed as

(1.2) dT dt ADE = β 0 T 4 ,
where β 0 is the heat effect coefficient. Then by the Fourier law, we derive the following law for heat transfer for the solar atmosphere:

(1.3) ∂T ∂t + (u • ∇)T = κ∆T + β 0 T 4 + β 1 (E 2 + H 2 ).
Here on the right-hand side, the first term represents the usual diffusion of heat, the last term is the heat source due to the solar electromagnetic fields. Importantly the second term represents the anti-diffusive effect of heat, and it is this anti-diffusive effect that leads to the formation of sunspots, the solar flares and the prominences.

Second, the full model governing Sun's surface plasma fluid combines the fluid dynamical equations, the above new heat equation (1.3), and the Maxwell equations. One key component of the theory for the formation of sunspots and solar eruptions is to prove a blow-up theorem, Theorem 3.1. This theorem shows that there exist x 0 ∈ Ω and t 0 > 0, such that the temperature T blows up at (x 0 , t 0 ) with blow-up time estimated as (1.4)

t 0 = |Ω| 3 3a 3 β 0 , a = Ω T 0 (x)dx,
where T 0 is the initial value of temperature, Ω = S 2 × (r 0 , r 1 ), r 0 is the solar radius, and r 1 = r 0 + h with the thickness of solar atmosphere h.

Third, the sunspots can now be clearly explained by the anti-diffusive effect of heat and the temperature blow-up that we just mentioned. We summarize this explanation as follows:

(1) Due to thermal fluctuations, the temperature in the solar atmosphere is nonhomogeneous, leading to elevated temperature in some local areas. The anti-diffusive effect of heat then shows that the higher temperature regions absorb more photons from their surrounding areas, leading to their temperature decreasing, and consequently generating sunspots; (2) The anti-diffusive effect of heat makes the temperature around the sunspot areas increasing rapidly, generating the temperature blow-up, and leading to solar eruptions. In fact, we deduce from the temperature blow-up at (x 0 , t 0 ) that

(1.5) lim t→t 0 du(x 0 , t) dt = ∞,
which represents the high speed gas explosion and particle ejections, with ejection direction given by

(1.6) r = -lim t→t 0 ∇T (x 0 , t) |∇T (x 0 , t)| .
(3) It is clear that the temperature blow-up generates solar flares. (4) By the Maxwell equations, it is clear that the eruption described by (1.5) generates very strong electromagnetic radiation in the r direction. (5) The eruption (1.5) leads also to a huge current jet J = ρ e u in the r direction, which, in view of the Ampère law, gives rise to violent magnetic loops, perpendicular to the direction r, leading to the solar prominences. (6) Astronomical observations show that sunspots and solar flares occur periodically in an 11-year cycle. The blow-up time (1.4) links the initial temperature T 0 , the solar eruption period, and the anti-diffusive effect coefficient β 0 . Such a link is applicable to all stars. For the Sun, we can easily estimate

β 0 = 2.85 × 10 -22 /(K 3 • s).
Fourth, the anti-diffusive effect of heat (1.1) may provide a new source of fuel. Basically, in a high temperature plasma system where free electrons are abundant, photons in the low temperature area experience the anti-diffusive effect and can move to the high temperature region, further raising the temperature in the high temperature region. The conversion of the thermal energy in the high-temperature region offers an alternative fuel source, which we call plasma fuel. The Sun is a natural source of photons for the low-temperature regions required for such a process to continue. This direct way of converting solar energy into a new form of fuel will be much more efficient than the photovoltaic devices.

In view of the non blow-up condition (3.30), the following condition is required for the anti-diffusive effect of heat to dominate the normal heat conduction:

(1.7) κ < 16 9 β 0 |Ω| 1/15 Ω T 5 0 dx 3/5
, where T 0 is the initial temperature, κ is the heat conduction coefficient, and β 0 is the anti-diffusive effect coefficient. Both κ and β 0 depend on the material. Therefore, for a given plasma system, the larger the domain and the higher the initial temperature, the stronger the antidiffusive effect. Consequently an effective plasma device for converting solar energy into a fuel source can be achieved by increasing the domain size and/or the initial temperature.

The paper is organized as follows. Section 2 introduces the antidiffusive effect of heat and the model for the Sun's surface fluid dynamics. Section 3 proves the blow-up theorem, derives a non blow-up condition, and provides the detailed explanation of the formation of solar surface eruptions, solar flares, and solar prominences.

Sun's Surface Fluid Dynamics

2.1. Astronomical phenomenon on the Sun's surface. The Sun is a star which we are most familiar with. The solar mass is about 2 × 10 30 kg, and the Sun's radius is 7 × 10 5 km. Its average density is ρ = 1.4g/cm 3 and, for comparison, we recall that the density of water is ρ = 1g/cm 3 . The Sun consists mainly of hydrogen (94%) and helium (6%), and is entirely in a gaseous state. Hence it can be viewed a gaseous fluid ball.

Solar structure is divided into two parts: the interior and the atmosphere. The atmosphere is composed of three distinct parts: the photosphere, the chromosphere and the corona. The photosphere is in the bottom layer, the chromosphere is in the middle and the corona is in the outer of the solar atmosphere. The most visible light come from the photosphere, and its temperature is ranged in 4000K ∼ 7000K. The chromospheric temperature is 1.5 × 10 4 K, and the corona has the highest temperature at about 2 × 10 6 K. But the corona density is very low, at 10 -9 times the density of the earth's atmosphere. Based on astronomical observations, the following are known important phenomena for the solar atmosphere:

1) Solar flare. It is a very attracting event which occurs in Sun's atmosphere, and is a sudden flash in Sun's surface brightness. Solar flares affect all three layers: photosphere, chromosphere, and corona, and are often accompanied by a coronal mass ejection, or by an erupting prominence. When the plasma medium is heated to near T = 5×10 6 K, flares are powered by the sudden release of huge energy, together with very strong electromagnetic radiations and high speed (near the speed of light) particle eruptions. The active period of solar flares follows the 11-year cycle, called the solar cycle. See Figure 2.1 for solar flares. 3) Sunspots. They are temporary phenomena on the Sun's surface that appear as spots darker than the surrounding areas. The sunspot regions have lower surface temperature. Sunspots occur in an approximatively 11-year solar cycle, the same as that of the solar flares.

Usually, sunspots accompany secondary phenomena such as coronal loops, prominences and solar flares. Most solar flares and coronal mass ejections originate in active regions around sunspots.

Astronomical observations show that solar flares, prominences and sunspots are intimately related. The theory that we established in this paper verifies clearly this claim, and explains their relations. 2.2. Anti-diffusive effect of heat. In [START_REF] Ma | Statistical theory of heat[END_REF], the authors developed a statistical theory of heat, consisting of the following main ingredients:

1). Energy level temperature formulas. These formulas are derived from the well-known Maxwell-Boltzmann, the Fermi-Dirac, and the Bose-Einstein distributions, and they show that the temperature is essentially the average energy level of system particles.

2). Photon number entropy formula. This formula shows that entropy is the number of photons in the gap between system particles, and the physical carrier of heat is the photons.

3). Vibratory mechanism of photon absorption and radiation. A particle can only absorb and radiate photons while experiencing vibratory motion. The higher the frequency of the vibration of the particle, the larger the absorbing and radiating energy. The vibration or irregular motion of the particles in a system is caused by collisions between particles and by absorbing and radiating photons. 4). Law of heat transfer. For particles in high speed vibration and irregular motion, the rate of photon emission and absorption increases, leading to the number density of photons to increase, and further causing the particle energy levels to elevate. Hence, the photon absorption and emission induce the concentration of the temperature, which we call the anti-diffusive effect of heat:

(2.1)
Due to the higher rate of photon absorption and emission of the particles with higher energy levels, the photon flux will move toward the higher temperature regions from the lower temperature regions.

On the other hand, we know that temperature obeys the Fourier law, i.e. photons are dictated by the Fick law. Hence, the heat transfer follows the balance between the anti-diffusive effect (2.1) and the Fick diffusion law for the system photons, i.e. the law of heat transfer can be expressed as (2.2) dT dt = κ∆T + Q + the concentration rate of photons, where κ∆T represents the diffusion term, Q is the heat resource.

Remark 2.1. In the classical heat conduction theory, the law of heat transfer is written as

(2.3) dT dt = κ∆T + Q, which is different from (2.
2). In fact, in the case where the temperature is not very high, or the system medium is not plasma, the third term in the right-hand side of (2.2) is very small and can be ignored. However, for the Sun's surface plasma fluid, this term will play a crucial role for the appearance of the solar flares.

Based on the anti-diffusive effect of heat (2.1), the concentration of photons is a reversed process of heat diffusion. Hence, by the Stefan-Boltzmann law, the third term on the right-hand side of (2.2) should be proportional to T 4 :

(2.4) the concentrating rate of photons = β 0 T 4 .

Equivalently the anti-diffusive effect of heat is expressed as

dT dt ADE = β 0 T 4 ,
where β 0 is the anti-diffusive effect coefficient.

In fact, it is the anti-diffusive effect of heat (2.1) that leads to the formation of sunspots, and it is the relation (2.4) that yields the solar flares and prominences. 2.3. Sun's surface fluid dynamical equations. Sun's surface fluid is composed of gaseous plasma. The state functions describing solar flares and prominences are: the velocity field u of the plasma fluid, the temperature T , the electromagnetic fields E and H. Hence, the dynamical model governing Sun's surface fluid is the three groups of equations coupling the Navier-Stokes equations, the heat equation, and the Maxwell equations.

We start with the spatial domain given by Ω = S 2 × (r 0 , r 1 ), where S 2 is the two-dimensional unit sphere, r 0 is the solar radius, and r 1 = r 0 + h with the thickness of solar atmosphere h. We bow describe the three set of equations, and their coupling.

1). The Navier-Stokes equations are written as

(2.5) ρ ∂u ∂t + (u • ∇)u = µ∆u -∇p + f,
where ρ is the mass density, p is the pressure, and f is the force density.

Because the fluid is plasma, each particle is charged. Hence, the force field f includes f = charge force + Lorentz force + thermal force.

Based on classical electromagnetic theory,

charge force = ρ e E, Lorentz force = J × H,
where ρ e is the effective charge density in the plasma, J = ρ e u is the effective current density. By theory of thermodynamics, we have

thermal force = -gk k kρ(1 -αT ),
where g is the solar gravitational constant, α is the thermal expression coefficient, and k k k is the radial unit vector. Then equations (2.5) are expressed as

(2.6) ρ ∂u ∂t + (u • ∇)u = µ∆u -∇p + ρ e E + ρ e u × H -g kρ(1 -αT ).
Also, (2.6) is complemented with the continuous equation (2.7) ∂ρ ∂t + div(ρu) = 0.

2). In view of (2.2) and (2.4), the heat equation is given by

(2.8) ∂T ∂t + (u • ∇)T = κ∆T + Q + β 0 T 4 ,
where κ is the heat conduction coefficient, and Q is the heat source excited by the solar electromagnetic fields, written as

Q = β 1 (E 2 + H 2 ).
Thus, the heat equation (2.8) becomes

(2.9) ∂T ∂t + (u • ∇)T = κ∆T + β 0 T 4 + β 1 (E 2 + H 2 ).
3). The Maxwell equations read (2.10)

µ 0 ∂H ∂t = -curl E, ε 0 ∂E ∂t = curl H -J, J = ρ e u, div H = 0, div E = ρ e ,
where µ 0 is the magnetic permeability, and ε 0 is the dielectric constant.

4). Model for Sun's surface plasma fluid. Combing the fluid dynamical equations (2.6)-(2.7), the heat equation (2.9), and the Maxwell equations (2.10), we derive the model governing Sun's surface plasma fluid as follows

ρ ∂u ∂t + (u • ∇)u = µ∆u -∇p (2.11) + ρ e (E + u × H) -gk k kρ(1 -αT ), ∂T ∂t + (u • ∇)T = κ∆T + β 0 T 4 + β 1 (E 2 + H 2 ), (2.12) ∂H ∂t = - 1 µ 0 curl E, (2.13) ∂E ∂t = 1 ε 0 curl H - 1 ε 0 ρ e u, (2.14) div H = 0, (2.15) div E = ρ e , (2.16) ∂ρ ∂t = -div(ρu). (2.17)
The system (2.11)-(2.17) constitutes the basis to establish Sun's electromagnetic eruption theory.

Theory on Formation of Sunspots and Solar Eruptions

3.1. Blow-up theorem. From the mathematical point of view, the phenomena of solar flares and prominences correspond to the blow-up of the solutions of the equations (2.11)-(2.17). Blow-up is mathematical property that a solution Φ(x, t) = (u, T, H, E) is called to blow-up at (x 0 , t 0 ) if (3.1) lim

t→t 0 |Φ(x 0 , t)| = ∞,
where

|Φ| 2 = u 2 + T 2 + E 2 + H 2 .
Hence, the blow-up theorem for the system (2.11)-(2.17) introduced in the following is crucial for us to understand the Sun's electromagnetic eruptions. For simplicity, consider the case where ρ is a constant, and (2.17) becomes

(3.2) divu = 0.
Consider the initial and boundary conditions:

u n | ∂Ω = 0, ∂u τ ∂n ∂Ω = 0, ∂T ∂n ∂Ω = 0, (3.3) Φ = (u, T, H, E)| t=0 = Φ 0 , (3.4)
where n, τ are the unit normal and tangent vectors on ∂Ω. Then we have the following theorem. sup

Ω |Φ 0 (x)| < ∞,
then there exist x 0 ∈ Ω and t 0 > 0, such that the temperature T blows up at (x 0 , t 0 ), and consequently Φ(x, t) blows-up (x 0 , t 0 ) as well, i.e. (3.1) holds true.

Proof. Take the integration on both sides of (2.12):

(3.5) d dt Ω T dx = Ω [κ∆T -(u • ∇)T + β 0 T 4 + β 1 (E 2 + H 2 )]dx.
In view of (3.2) and (3.3), by the Gauss formula, we have

Ω ∆T dx = ∂Ω ∂T ∂n dx = 0, Ω (u • ∇)T dx = - Ω T divudx + ∂Ω T u n dS = 0.
Then (3.5) becomes

(3.6) d dt Ω T dx = Ω [β 0 T 4 + β 1 (E 2 + H 2 )]dx.
Physically, T > 0, which can also be proved using the same method as in the paper [START_REF] Foias | Attractors for the Bénard problem: existence and physical bounds on their fractal dimension[END_REF]. Hence, we have

(3.7) Ω |T |dx = Ω T dx > 0.
In addition, by the anti-Hölder inequality (see [START_REF] Ma | Theory and Methods of Partial Differential Equations[END_REF]), for any 0 < p < 1 and q = p/(p -1) we have

(3.8) Ω |f g|dx ≥ Ω |f | p dx 1/p Ω |g| q dx 1/q .
Take f = T 4 , g = 1, p = 1/4 and q = -1/3, and note (3.7), then (3.8) becomes

Ω T 4 dx ≥ 1 |Ω| 3 Ω T dx 4 . (3.9)
Recall the comparison theorem of differential equations: For two non-negative functions f 1 and f 2 satisfying

f 1 (t) ≤ f 2 (t), ∀ t ≥ 0, we consider the initial value problems dx 1 dt = f 1 (t) with x 1 (0) = a, dx 2 dt = f 2 (t) with x 2 (0) = a,
where a ≥ 0. Their solutions x 1 (t) and x 2 (t) satisfy

x 1 (t) ≤ x 2 (t), ∀ t ≥ 0.
Then, consider the equation

d dt Ω T dx = β 0 |Ω| 3 Ω T dx 4 . (3.10)
Based on the comparison theorem, for the solution Ω T dx of (3.6) and the solution Ω T 1 dx of (3.10) with the initial value conditions

Ω T dx t=0 = a, (3.11) Ω T 1 dx t=0 = a, (3.12
) we deduce from (3.9) that where k = 1/|Ω| 3 . It is easy to see that the solution of (3.14) is

Ω T 1 dx ≤ Ω T dx, ∀ t ≥ 0. ( 3
(3.15) y = a [1 -3ka 3 β 0 t] 1/3 , k = 1/|Ω| 3 .
Again, by (3.13) we have,

(3.16) y ≤ Ω T dx,
where T is the solution of (2.12). By (3.15), we see that

lim t→t 0 y(t) = ∞, t 0 = 1 3ka 3 β 0 .
Then we deduce from (3.16) that (3.17) lim

t→t 0 Ω T dx = ∞,
which implies this theorem holds true.

3.2. Sun's surface eruptions and sunspots. The blow-up theorem, Theorem 3.1, provides a solid mathematical foundation for the solar eruption theory developed in this paper. It shows that the eruptions are typically topological phase transitions at the blow-up point t 0 = 1/3ka 3 β 0 , at which the state functions Φ(x, t) = (u, T, H, E) tend to infinite, depicting the huge and complicated explosions as high speed mass ejections, sudden flares of flashlight, very strong radiations, and large spurts of loop shaped magnetic energy.

Based on Theorem 3.1 and the anti-diffusive effect of heat (2.1), we discuss the sunspot and solar eruption problems in the following. 1). Sunspots. We know that sunspots are the regions on Sun's surface, and possess the following two main characteristics:

(1) the temperature in sport is lower than the surrounding areas, and consequently the brightness is darker; and (2) it often accompanies secondary events as the solar flares and prominences -the solar eruptions. The above two characteristics can be well explained as follows by the anti-diffusive effect of heat (2.1) and the blow-up of the temperature:

(1) Due to thermal fluctuations, the temperature in the solar atmosphere is nonhomogeneous, leading to elevated temperature in some local places. Based on the anti-diffusive effect, the regions with higher temperature absorb more photons from their surrounding areas, leading to the temperature decreasing, and consequently generating the sunspots;

(2) When the sunspots appear, the anti-diffusive effect (2.4) makes the temperature in the regions around sunspots rapidly increasing to generate the blow-up as described in Theorem 3.1, which causes solar eruptions.

2). Solar flares. In the proof of Theorem 3.1, we derived (3.17), which implies that there exists a point x 0 ∈ Ω such that (3.18) lim

t→t 0 T (x 0 , t) = ∞ with t 0 = 1 3ka 3 β 0 .
It is the blow-up (3.18) that generates the solar flare. In addition, it also induces a chain of tragical variations in the velocity field u and the electromagnetic fields H and E, resulting in the coronal mass ejections, the strong radiations, and the solar prominences.

3). Coronal mass ejections. Equation (2.11) dictates the behavior of mass ejections. When the temperature T blows up at (x 0 , t 0 ) as given in (3.18), the maximal forces acting on the particles near x 0 are just ∇p, as 0 < αT < 1 in (2.11). Hence, in the neighborhood of (x 0 , t 0 ), (2.11)

∂ 2 E ∂t 2 + 1 ε 0 µ 0 curl 2 E = - ρ e ε 0 ∂u ∂t , ∂ 2 H ∂t 2 + 1 ε 0 µ 0 curl 2 H = ρ e ε 0 u.
In view of ( , where β 0 represents the heat effect coefficient, depending on the material, and T 0 is the initial average temperature. The period formula (3.28) is also applicable to all stars, because it was found that other stars also have the eruption phenomena. For the Sun, t p = 11 years = 3.469 × 10 8 s, and T 0 is the average temperature of the chromosphere:

Ω T dx = |Ω|T 0 [1 -3T 3 0 β 0 t]
T 0 = 1.5 × 10 4 K.
Therefore, the heat effect coefficient for solar gaseous plasma is derived from (3.28) as follows

(3.29) β 0 = 2.85 × 10 -22 /(K 3 • s).
Remark 3.2. The heat effect coefficients for non-plasma materials are much smaller than the value in (3.29). , where T 0 is the initial temperature, κ is the heat conduction coefficient, and β 0 is the heat effect coefficient. Note that β 0 is negligible for nonplasma material. Therefore, in the usual cases we often encounter, both T 0 and β 0 are small, so that no temperature blow-up occurs.

To verify (3.30), we consider the heat conduction equation with antidiffusive effect of heat: By the Sobolev inequality (see e.g. p.23 in [START_REF] Ma | Theory and Methods of Partial Differential Equations[END_REF]):

Ω |∇T | 2 dx 1/2 ≥ n(n -2) 2(n -1) Ω T 2n n-2 dx n-2 2n
, where n is the dimension of space 2 . In this paper n = 3, and we have . 2 We remark here that by the above inequality and the Hölder inequality, we have ∇u L p ≥ c |Ω| (N * -q)/N * q u L q for n > p, N * = np n -p . Then, (3.38) implies that (3.37) holds true, and consequently the heat system (3.31) has no blow-up. Physically, (3.36) means that (3.30) is the condition to ensure no temperature blow-up.

Figure 2 . 1 .

 21 Figure 2.1. Solar flares

Figure 2 . 2

 22 Figure 2.2

Theorem 3 . 1 (

 31 Blow-up theorem). Let Φ(x, t) be a solution of (2.11)-(2.16) and (3.2), and satisfy the initial and boundary conditions (3.3) and (3.4). If the initial value Φ 0 is bounded, i.e.

  .13) Denote y = Ω T 1 dx, then equation (3.10) with the initial value condition (3.12) is written as (3.14) dy dt = kβ 0 y 4 , y(0) = a,

( 3

 3 .24) curl H = ρ e u. By (3.21) and (3.22), we have (3.25) ρ e u • r 1 at (x 0 , t 0 ), which represents the huge current jet in the r direction.Then, by the Ampère law (3.24), the strong current of (3.25) gives rise to violent magnetic loops, perpendicular to the direction r in (3.22), as shown in Figure3.1. As the current orientation r is not in the normal direction of solar surface, the erupting magnetic loops are the solar prominences that we observe.

Figure 3 . 1 .

 31 Figure 3.1. Huge current jet J generates a violent magnetic loop, which is the solar prominence.

3. 3 .

 3 Period of solar eruptions. Astronomical observations show that sunspots and solar flares occur periodically in an 11-year cycle. The temperature solution (3.15) provides also the formula of solar eruption period, which is rewritten as(3.26) Ω T dx = a [1 -3ka 3 β 0 t] 1/3 ,where k = 1/|Ω| 3 , and a is the initial valuea = Ω T (x, 0)dx.When a solar eruption ends, the temperature distribution in the solar chromosphere is almost homogenous. Let T 0 be the homogenous temperature, i.e. T (x, 0) = T 0 . Then the initial value a = T 0 |Ω|, and (3.26) becomes(3.27)

3. 4 .

 4 Non blow-up condition of temperature. In this section, we demonstrate that under the Dirichlet boundary condition for the temperature T 1 , the non blow-up condition for temperature in domain

( 3 3

 33 .31)∂T ∂t = κ∆T + β 0 T 4 , x ∈ Ω ⊂ R 3 , T | ∂Ω = 0.Multiplying both sides of (3.31) by T and taking integral, we have(| 2 + β 0 T 5 ]dx.

3 .

 3 In addition, by the Hölder inequality, we derive||T || L 6 ≥ 1 |Ω| 1/30 ||T || L 5 .

Then we deduce from ( 3 5 + 5 dx 2 / 5 ≥ C 0 ΩT 2

 3552502 ε, ∀ t > 0,where ε > 0 is arbitrarily small, then we have (3.37) lim t→∞ Ω T 2 dx = 0. In fact, by the inequality Ω T dx , and by the Gronwall inequality, we deduce from (3.35) that (3.38) Ω T 2 dx ≤ e -C 0 t 0 α(τ )dτ ,

  3.21) and(3.22), the equations (3.23) generate very strong electromagnetic radiation in the r direction. 5). Solar prominences. Consider the equation (2.14), which can be approximatively expressed as (i.e. the Ampère law):

  1/3 . It is easy to deduce from (3.27) that the time t p for the next eruption

	to occur satisfies
		1 -3T 3 0 β 0 t p = 0.
	Hence the period t p is
	(3.28)	t p =	1 3T 3 0 β 0

The Dirichlet boundary condition for T amounts to saying that there is heat exchange of the system with outside. Hence there is no contradiction between the non blow-up condition and the blow-up theorem 3.1, where the Neumann boundary condition for T is used.

The work was supported in part by the US National Science Foundation (NSF), the Office of Naval Research (ONR), and by the Chinese National Science Foundation.

3.3.


Mario Barbareschi

Alberto Bosio

Hoang Anh

Said Hamdioui Du Nguyen

Said Hamdioui

Marcello Traiola

Elena Ioana Vatajelu

Memristive

Memristive devices: Technology, Design Automation and Computing Frontiers

Keywords: component, emerging technologies, computer architecture, memory, CAD tool, logic synthesis. I

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Today's computing devices are based on the CMO S technology, that is the subject of the famous Moore's Law [1], predicting that the number of transistors in an integrated circuit will be doubled every two years. Despite the advantages of the technology shrinking, we are facing the physical limits of CMO S. Among the multiple challenges arising from technology nodes lower than 20 nm, we can highlight the high leakage current (i.e., high static power consumption), reduced performance gain, reduced reliability, complex manufacturing process leading to low yield, complex testing process, and extremely costly masks [START_REF] Hoeftlinger | Chips2020:AGuidetotheFutureotNanoelectronics[END_REF] [3] [4] [START_REF] Gielen | Emergingyieldandreliabilitychallengesinnanometer CMOS technologies[END_REF].

Additionally, the expected never-ending increasing of performances is indeed no longer true. Looking in more detail, the classical computer architectures, either Von Neumann or Harvard, divide the computational unit (i.e., CPU) from the storage element (i.e., memory). Therefore, data have to be transferred inside the computational element in order to be processed and then transferred back to be stored. The main problem of this paradigm is the bottleneck due to the data transfer time limited by the bandwidth. For instance, transferring one TeraByte at the rate of 1 Gbit/second re q uires more than two hours.

Many new technologies are under investigation, among them the memristor is a promising one [START_REF] Chua | Memristor-themissingcircuitelement[END_REF]. Indeed, the memristor is a non-volatile device able to act as both storage and information processing unit that presents many advantages: CMO S process compatibility, lower cost, zero standby power, nanosecond switching speed, great scalability, high density and non-volatile capability [START_REF] Waser | Redox-based resistive switching memories-nanoionic mechanisms, prospects, and challenges[END_REF] [START_REF] Yang | Memristive devices for computing[END_REF]. Thanks to its nature (i.e., computational as well as storage element), the memristor is exploited in different kind of applications, such as neuro morphic systems [START_REF] Burger | Variation-tolerant computing with memristive reser voirs[END_REF], non-volatile memories [START_REF] Kim | A functional hybrid memristor crossbar-array/cmos system for data storage and neuromorphic applications[END_REF], computing architecture for data-intensive applications [START_REF] Hamdioui | Memristor based computation-in-memory architec ture for data-intensive applications[END_REF].

This paper presents a comprehensive overview of memristor technology and its potential to design a new computational paradigm. The remainder of the paper is structured as following. Section 11 presents the basic background about the memristor and its potential. Section III discusses the design flow of memristor-based computing devices by presenting a synthesis flow and the design exploration framework. Finally, Section 0 discusses the real impact of memristor-based computing devices.

II.

MEMRlSTIVE DEVICES AND THEIR POTENTIAL

The continuous technology scaling, as well as the emergence of new technologies, favor increasing of the system complexity and performance, opening the scientific community to exotic applications and computation paradigms which had been unfeasible a few years back due to technological limitations of the hardware. The emergence of new low power, highly scalable, CMO S compatible memory devices (such as memristive devices) is tries to address the technical constraints of today's memories.

The memristive devices have great characteristics in terms of area, power and speed when used as memory or data storage devices, but, in addition, they are promising solutions for logic implementation. Thanks to the relative easiness of massive parallelism, computing in memristive memory becomes trending topic in current research activities. Moreover, other fields, such as brain-inspired computing, benefit from the uni q ue features of this technology. In continuation, a comprehensive overview of the memristive technology landscape with special emphasis on the most desirable features and dire shortcomings in memory and logic design is presented.

A. Working Principle and Classification

The memristor is a semiconductor device whose resistance is called memristance. The memristance is a charge dependent resistance and its value varies as a function of current and flux. The memristor technology has some great advantages such as data non-volatility, CMO S compatibility, low switching power, no leakage power, high integration capability.

Memristors can be classified in two different types: (i) ionic thin film and molecular memristors, and (ii) magnetic and spin based memristors. In this work we are focusing on the first category, since the second has developed independently as spintronic devices.

When used as a memory device, the first category of memristors is called resistive memory, more precisely Resistive Random Access Memory (RRAM) and it can act as a non-volatile memory. Its data storage element is a three-layer device, consisting of a dielectric sandwiched between two metal electrodes. There are many materials which can be used for thc clcctrodcs and diclcctric, but thc undcrling opcration principlc rcmains thc samc. Thc RRAM dcvicc switchcs between two resistive states, i.e., the high resistance state (HRS) and the low resistance state (LRS), when triggered by an electrical input.

RRAM relies on the formation (corresponding to low resistance) and the rupture (corresponding to high resistance) of conductivc paths in thc diclcctric laycr. Oncc thc conduction path is formed, it may be RESET (the path broken, transition ii'om LRS to HRS) or SET (the path re-formed, transition from HR S to LRS). Usually, right after fabrication (i.e., the pristine samples) the devices have a very high electrical resistance (-1 GQ) and a large voltage is required for the first SET operation, also known as the forming process; this drastically reduces the device resistance (to about IOKQ) triggering the switching behavior in the subsequent cycles.

Classification: The memristive devices (RRAM) can be classitied following ditTerent criteria. They can be c1assitied according to the used materials, the switching mechanism, the conductive path, and the switching mode. There are two possible RRAM switching modes: unipolar switching, which depends only on the amplitude of the applied voltage and not its polarity, i.e., the SET and the RESET opcrations arc controllcd by thc samc polarity; and bipolar switching in which thc SET and thc RESET opcrations arc controlled by reverse polarities.

Depending on the do minant physical switching mechanism, the resistive devices can be elassified in: Phase Change Memories, Electrostatic/ Electronic Effects Memories, and Rcdox Mcmorics. Va rious rcsistivc switching mcchanisms havc bccn proposcd to cfficicntly pcrform thc SET and RESET operations. They include the formation and rupture of conductive paths, charge trapping, electrode-limited conduction [START_REF] Lee | Accurate analysis of conduction and resistive-switching mechanisms in double-layered resistive-switching memory devices[END_REF], [START_REF] Zhu | An overview of materials issues in resistive random access memory[END_REF]. Thc low-rcsistancc conductive path can bc cithcr localizcd (filamcntary) or homogcncous.

One of the most versatile resistive memories is the Redox RAM [1], [START_REF] Yu | Overview of resistive switching memory (RRAM) switching mechanism and device modeling[END_REF], where the RESET and SET processes, breakdown and regrow of the conductive filaments, involve oxidation and reduction (i.e., redox reaction). These are Metal Insulator-Metal (MIM) structures, in which the switching mechanism is electrochemical and it can occur in the insulator-layer, or at the insulator-layer/metal contact interfaces.

Figure 1. Resistive device classification

The MIM structures can be classified by their underlying switching mechanism as follows [START_REF] Vatajelu | Nonvolatile memories: Present and future challenges[END_REF]:

The Valence Change Mechanism (VCM): here the dielectric layer can act as an electrolyte and the migration of oxygen vacancies within the applied electric field evolves in a bipolar manner. The conductive path is formed due to the oxygcn anions (positively chargcd oxygcn vacancics), while the electric current is defined by the electrostatic barrier in the band diagram. Appling negative bias voltage on the electrodes of the memristor the SET operation is performed due to a local redox reaction which increases the device conductivity. The RESET operation is performed by reversing the bias polarity and allowing thc rccombination of oxygcn. Thc most common examples of VCM RRAMs use TaOx, HfOx and TiOx [START_REF] Lim | Conduction Mechanism of Valence Change Resistive Switching Memory: A Survey[END_REF], [START_REF] Wedig | Nanoscale cation motion in TaOx, HfDx and TiOx memristive systems[END_REF] devices.

The Electrochemical Mechanism (ECM): uscs an clectrochcmically activc clectrodc mctal such as Ag or Cu. The mobile metal cations drift in the ion conducting layer and discharge at the counter-electrode, leading to a growth of conductive metallic filaments in the isolation layer -i.e., the SET mechanism. The RE SET mechanism is performed by reversing the polarity of the applied voltage, resulting in the electrochemical dissolution of the conductive filaments [START_REF]Electrochemical and thermochemical memories[END_REF].

The Thermochemical Mechanism (TCM): relies on a filament modification due to Joule heating. Conductive filaments, composed of the electrode metal transported into the insulator, are formed during the forming process prior to memory cyclic switching. The SET operation is achieved by Joule heating; it triggers local redox reactions that facilitate the formation of oxygen deficient ions and metallic filaments. The RESET operation is a thermally activated process resulting in a local decrease of the metallic species. TCMs are unipolar switching devices. N iO has emerged as the reference material for resistive switching based on the TCM [19]. As a reasonably representative example, in the subsequent sections the focus will be on HfOx-based VCM RRAMs, as they see:n the most promising. Note that the focus of the paper is on device test where the quality of the conductive path formation is relevant, regardless of the physical mechanism.

B. Opportunities and Challenges

Opportunities: Memristive are on the way to change the classical memory/storage architectures. They should meet the high demands of tomorrow applications, like high performance and high density, good endurance, small devices sizes, good integration, low power profile, resistance to radiation, and ability to scale below 20 nm 120 J, 1211.

The most investigated use of the memristor is memory since a it can store data. When compared with traditional memories, such as SRAM or DRAM, this kind of memory has many benefits, such as, no leakage power, non-volatility and scalability, being in the same time superior to flash memory in terms of speed and scalability. In addition, the memristive device can be used in logic circuits, either as standalone logic gates, or used in hybrid CMOS-memristor circuits. Memristors can be used to do digital logic using implication instead of NAND.

The simple device structure (metal-insulator-metal) of a RRAM device, its compatibility with CMOS process, the scaling opportunities below 8nm, its large on/off ratio, and fast operating speed make the RRAM devices ideal candidates to eventually be used as embedded memories.

Challenges: Amongst the greatest challenges faced by today's RRAM devices is their relatively low endurance (10 5 -10 10 cycles [22]) and poor uniformity. The low endurance limits their efficiency as embedded memories, while the poor uniformity causes extreme variability and limited reproducibility.

Another challenge is the large number of new materials (and combinations of materials) which can be used for the resistive stack formation (as seen in Fig. I a) making difficult the standardization of the fabrication process. The introduction of new materials in RRAM fabrication does not allow enough time to collect and generate the data required to guarantee sufficient yield. These issues, which are common to all emerging technologies, introduce aggressive challenges on defect and fault modelling and possible test solutions.

The main concern regarding the RRAM is the variability of its switching parameters. It has been demonstrated that lowering the switching power of RRAM will induce large variability of the filament formation. This is due to movement of only countable number of atoms [1]. This movement can cause device to device resistance variation but also cycle to cycle variation [START_REF] Chen | Variability of resistive switching memories and its impact on crossbar array performance[END_REF]. This has resulted in the need for enhancing the read/write power consumption in RRAM [START_REF] Xue | A 0.13 urn 8 Mb Logic-Based CuxSiyO ReRAM With Self-Adaptive Operation for Yield Enhancement and Power Reduction[END_REF] along with designing adaptive sensing circuits to mitigate this effect [START_REF] Park | Analysis of resistance variations and variance-aware read circuit for cross-point ReRAM[END_REF]. Moreover, it has motivated the utilization of other types of RRAMs such as the conductive bridging RAM (CBRAM) which has bigger high to low resistance ratio and non-filamentary RRAM devices in which the ions drift across the whole aperture of RRAM [START_REF] Govoreanu | A-VMCO: A novel forming-tree, selt:rectifYing, analog memory cell with low-current operation, non filamentary switching and excellent variability[END_REF].

In terms of scaling RRAM has shown to be promising device as its data storage is based on atomic movement. Theoretically, RRAM can scale down to size of a conductive filament and scaled devices down to 2nm has been reported. Nonetheless extensive scaling can make the filament so small and might induce retention problems [START_REF] Chen | Improvement of data retention in HfD2/Hf 1 TlR RRAM cell under low operating current[END_REF].

Note that as a result of filamentary switching, RRAM scaling will not be accompanied by scaling of the operating voltage and currents because the filament conduction is governed by the electrical programming conditions, therefore to achieve low power constrains proper material selection and optimized programming conditions is required [START_REF] Wouters | Phase-Change and Redox Based Resistive Switching Memories[END_REF].

From reliability perspectives, RRAMs have recently improved a lot from their early stage appearances. Their endurance cycles have increased from 10 3 up to 10 12 cycles and there has been some attempts to remove the initial forming step which is one of the sources for their resistance variability [START_REF] Wong | Metal-Oxide RRAM[END_REF]. A consolidation of material engineering along with optimizing the device operating parameters and novel techniques at circuit level are under research to further improve their reliability.

Due to attractive potential scaling and fast operation properties of RRAM they are considered as strong replacement for flash memories. Several prototyped chips have been presented for RRAM devices. As examples Panasonic has presented an 8Mb fast RRAM memory [START_REF] Kawahara | An 8 Mb Multi-Layered Cross-Point ReRAM Macro With 443 MB/s Write Throughput[END_REF], also Scandisk/Toshiba have presented a 32 Gb chip for high density applications.

TIT. TOWARD AUTOMA nON OF MEMRISTIVE DEVICE BASED CIRCUIT DESIGN

A fundamental component of any kind of computing architecture is the implementation of boolean logic functions thus, an automated tool for the synthesis of memristor-based circuits is mandatory [START_REF] Du Nguyen | Synthesizing HDL to memristor technology: A generic framework[END_REF][START_REF] Jintao | Skeleton-based design and simulation flow for Computation-in-Memory architectures[END_REF]. Tn [START_REF] Xie | Fast Boolean Logic Mapped on Memristor Crossbar[END_REF], the authors proposed a methodology for the synthesis of boolean logic function on a memristor-based crossbar. Their work showed that is possible to implement any kind of boolean function on a memristor based crossbar. Tn [START_REF] Traiola | XbarGen: A memristor based boolean logic synthesis tool[END_REF], we illustrated a methodology to automatically map an arbitrary boolean function to a memristor-based crossbar implementation. By applying different minimization tools and different synthesis parameters, we also showed that each obtained architecture is strongly dependent on them. Design Space Exploration (DSE) is therefore mandatory to help and guide the designer to select the best architecture.

Bearing in mind such consideration, in this section, we present a formal DSE approach that aims to calculate interesting circuits attributes avoiding simulation campaigns. We propose an algorithmic method to estimate both workload independent attributes (e.g. performance, area, etc.) and work load dependent ones. In particular, we estimate the power consumption of a given memristor-based crossbar architecture (the Fast Boolean Logic Circuit [START_REF] Xie | Fast Boolean Logic Mapped on Memristor Crossbar[END_REF]) providing both a lower and an upper bound for the power consumption and an error estimation.

A. Synthesis Flow and DSE

As described in [START_REF] Xie | Fast Boolean Logic Mapped on Memristor Crossbar[END_REF], FBLC approach implements a boolean function as a Sum-of-Product (SoP). Thus, the resulting crossbar has to be configured accordingly to the function's minterms. The proposed synthesis flow is depicted in the OThe input of the flow is the target boolean function that is minimized by using two different synthesis tools (i.e., ABC [START_REF][END_REF] and SIS [START_REF] Sentovich | SIS: A System for Sequential Circuit Syn thesis[END_REF]). Actually, we exploited two different tools to estimate the impact of different synthesis parameters and algorithms on the circuit characteristics (i.e., performance, area, power consumption, etc.). More in detail, SIS is employed for generating 2-levels logical networks while ABC is exploited for generating multi-levels logical networks. The result is the boolean function minimized and described as a set of minterms. As described above, different descriptions can be obtained. The subsequent step is the mapping of the minimized boolean function onto a crossbar-based memristor circuit. The tool XbarGen [START_REF] Traiola | XbarGen: A memristor based boolean logic synthesis tool[END_REF] can extract the function's minterms from the generated representation in order to analyze them and to build the corresponding FBLC circuit. The result is the set of VHDL files modelling the crossbar circuit. Finally, the crossbar VHDL model can be simulated by using any available logic simulator.

During the mapping process, XbarGen extracts the crossbar attributes that will be exploited by the proposed formal DSE approach. Let us first detail those attributes before moving to the DSE description. They can be divided in two main categories, namely the workload independent and workload dependent. Next subsections describe both of them and last subsection details the formal DSE.

B. Workload independent attributes

The workload independent attributes do not need any sim ulation (i.e., we do not have to simulate the crossbar VHDL model) to be evaluated. They are extracted by XbarGen during the mapping process and they are formalized as follows: Number of memristors in the circuit defined by the following equation: given that:

• Indexes i and ) run on minterms and crossbars respectively;

• Nin and Noul are the number of inputs and outputs respectively;

• Nocc ( mi , � ) is the number of occurrence of i-th minterm in j-th crossbar;

• NUl (m, ) is the number of literals of i-th minterm;

• Pii is equal to 1 if the i-th minterm is present in the )-th crossbar, otherwise it is equal to 0;

• Tc is the 'Latency' of a Crossbar;

• Ne is the Number of Crossbars in the circuit.

C. Workload dependent attributes

The workload dependent attributes require the simulation of the generated VHDL circuits to be evaluated. Tn this work, we consider the power consumption as workload dependent attributes formalized as: )-th crossbar that switch from '0' to '1' and from '1' to '0' respectively;

P = L [NuP j . Cup + Ndown j . Cdown]
• Cup and Cduwn are the power consumption of a memristor switching from '0' to '1' and from ' l' to '0' respectively.

It is worth to note that N,pj and Ndownj depend on the applied workload.

D. Formal DSE

The main goal of the proposed DSE is the characterization of the synthesized crossbars w.r.t. the above identified attributes. The idea is to avoid any simulation to speed up the DSE. Clearly, for the workload independent attributes the formal DSE is straightforward since it is enough to exploit the equations (1), ( 2) and (3). Thc challcnging issuc is dctcrmining thc actual powcr consumption. Even if the power consumption is a workload dependent attribute, we will show how to compute two bounds that cannot be exceeded by the actual power consumption: a worst case bound and a best case bound. It is wOlih to emphasize that such bounds will be computed without any simulation.

Referring to the equation ( 4), the idea that we exploit is to identify within the crossbar the elements that do not depend on actual inputs and manage those which are dependent on actual inputs. Thus, we can observe -as 0 shows -that the architecture has a first RE SET stage (INA) in which all the memristor in the circuit are set to ' 1 ' . Therefore, during this stage, we have only the contribution of N"pj * Cup while, during the rest of the computation, only NdolVnj * CdolVn contributes to the power consumption. Moreover, in both worst and best case scenarios, we consider a concatenation of executions providing inputs which trigger the worst and the best case respectively. Bearing in mind this, we can observe that, whether Ndownj memristors switch from ' 1 ' to '0' during the computation, the RESET phase has to switch the same number of memristors from '0' to '1'. Therefore, considering both worst and best case, we can assume that the two contributions are equal: In ordcr to cstimatc this contribution, let us considcr that a crossbar can be divided in 4 parts, as detailed in [START_REF] Traiola | XbarGen: A memristor based boolean logic synthesis tool[END_REF]. Hence we can assume the following:

• Concerning the green IN box, it is clear that half of the input memristors are going to switch during each execution of the circuit.

•

The same consideration is true for the output memristors, in the blue OL box.

Therefore, such two blocks of mernristors can be evaluated, in terms of switching memristors, independently from the actual input values. Thus, we are able to rewrite the equation 5 as follows: whcrc:

• Nintj is the number of mernristors that belong to the minterm boxes NAND and AND within the j-th crossbar that switches during the computation:

(7)
Let us now discuss about the remaining orange NAND and the red AND boxes. The memristors of these parts switch accordingly with the actual input values. For them, the goal is to find the two bounds. It is worth highlighting that the number of switching memristors in the AND box depends on which memristors switch in the NAND box. Therefore, the best and worst cases are computed by considering only the NAND box. Bearing in mind that half of the input memristors will eventually switch 1 ---+ 0 during each execution of the circuit, to find the best and worst input vectors we count, for each vertical nanowire, the number of memristors in the NAND box. For each couple of literals xi and xi (vertical nanowires) we consider, as for the best case, the one that leads to the lowest number of memristors and, as for the worst case, the one that leads to the biggest one.

Finally, we compute how many AND memristors will switch using the selected input vectors: since the minterms box is performing a NAND operation, if a minterm has at least one literal among those in the selected input vectors, the corresponding memristor in the AND box will not switch, otherwise it will.

As can be seen, we do not need the truth table of the function for retrieving (NmNAND , Nm/1ND)worstibesb indeed we only count the number of memristors in the NAND box in order to find the best and the worst combinations of inputs and then we verify if each minterm will be whether '0' or '1'. Therefore, the algorithm has a linear complexity. This is a grcat improvcmcnt comparcd to doing a simulation with all thc combinations of inputs that would lead to a complexity 8(2n), with n the number of inputs

E. Experimental Results

This section provides experimental results achieved by the proposcd flow. A bunch of combinatorial circuits arc uscd as bcnchmarks, dctails about circuits charactcristics arc available in [START_REF] Yang | Logic Synthesis and Optimization Benchmarks User Guide[END_REF]. Table I reports the achieved results for multiple crossbars. For each circuit, tables report the number of inputs (lN), minterms, memristors (Nm), the area, the estimated power consumption from PIVors! downto Pbes! including the error estimations (EIVors! and Ebes!). Moreover, in columns DSE Time and SimTime Overhead we report respectively the execution time of the proposed formal DSE and the overhead of a single simulation of the synthesized VHDL circuit compared to DSE. As shown in the tables, the cost of performing a full simulation to determine the power consumption is in average very high: about 300% for the multiple crossbar. This clearly prove the benefits of using our approach instead of a full simulation.

It is worth mentioning that, on a given circuit, the simulation is performed on a single workload while the formal DSE execution is actually independent from any specific workloads. Thus, the formal DSE takes only an execution in order to perform the estimation of both best and worst cases of power consumption, along with an error estimation. The simulation approach, on the other hand, require a full simulation per each workload to find actual best and worst cases of power consumption. Memristive devices are under investigation to explore not only their potential for building memories and logic, but also to build radically new computing architectures [START_REF] Hamdioui | Memristor For Computing: Myth or Reality?[END_REF], such as computing-in-memory (CIM).

CIM concept as proposed in [START_REF] Hamdioui | Memristor based computation-in-memory architec ture for data-intensive applications[END_REF] is based on the tight integration of computation and storage in a very dense crossbar array where memristors are injected at each junction of the crossbar (top electrode and bottom electrode). The communication and control from/to the crossbar can be realized using CMO S technology; it dictates what kind of operations have to be performed within the crossbar depending on the sequence and the voltages applied to the wordlines and the bitlines of the crossbar. The integration of CIM die with a conventional CPU can enable the realization of accelerators for specific application; these accelerators could outperform the traditional CPU with orders of magnitude [START_REF] Hamdioui | Memristor based computation-in-memory architec ture for data-intensive applications[END_REF].

In the rest of this section, we will briefly discuss an example of such an accelerator, including the architecture and principal of working, the potential applications that could benefit from such architecture, and some preliminary performance results.

A. Principal a/working Figure 4 shows the CIM-based computing architecture concept. It includes a conventional CPU, CIM die, main memory DRAM and external memory. In a conventional architecture, the CPU fetches, decodes and executes a big data program in which intensive memory accesses costs enormous energy consumption and significantly degrades the overall performance. In CIM-based architecture, the aim is to have as much as possible parts of the program (requiring big data) executed locally within the CIM-die; i.e., instead of moving data back and forth to the cache & register file, the (big) data which is initially stored in the CIM die, will be kept in the same location and (ideally) all parts of the program that should be executed on this data should take place locally within the CIM-die. Only the final result is moved from the CIM-die to the master CPU. Note that the operations within CIM-die are performed within the (non-volatile) memory; moreover, the CIM die can perform multiple operations simultaneously without the overhead of fetching data from memory [START_REF] Hamdioui | Memristor based computation-in-memory architec ture for data-intensive applications[END_REF][START_REF] Du Nguyen | Computation-In-Memory Based Parallel Adder[END_REF]. Therefore, significant reduction in memory access and the use of non-volatile technology for the CIM-die will not only improve the overall performance, but also dramatically reduce energy consumption. Note that idea is to have the (big) data (on which the CIM die needs to perform the operations) loaded to the die itself, and that the capacity of the memory within the die is large enough to store this data. In addition, the optimized performance is obtained in case the we have different operation applied on the same data, which is not changing frequently; this benefit also the endurance of the non-volatile memory of the CIM die. It is well recognized that the communication between the CPU and the memory is the killer for both performance and energy, especially for data-intensive applications; loading data from on chip SRAM and from off-chip DRAM cost about 50X, respectively, 6400X more energy as compared with an ALU operation [START_REF] Brunhave | [END_REF]. Clearly reducing the communication will have a significant impact not only on the energy consumption but also on the overall performance. Data-intensive applications which requires the execution of algorithms containing also operations on big data could benefit from the architecture discussed above. The big data can be stored on the CIM die and all the operations that have to used such data can be execute locally with the CIM die, by have the CPU provide appropriate instruction to the local CIM controller; hence, no need for data movement. Moreover, the CIM dies can perform its operation while the CPU is also executing other operation in parallel, resulting in overall performance improvement.

To illustrate the above, assume the program of Figure 5. The program has three loops that need to make use of the big data, and are supposed to run on the CIM die; obviously the data should be stored on the same die. Each time the loop is invoked, the CPU sends a request to the CIM die; the latter, performs the requested operations and returns the results to the CPU. It is possible to maximize the overall performance by having the CPU send the instruction to the CIM die ahead of time and such that the CIM-die will make sure that the results are ready when needed by the CPU. Examples of applications that have similar characteristics as the program of Figure 5 are database applications, where multiple queries (each consisting of large loops) are applied to a fixed database. These queries are used to look for specific data patterns in the database.

C. Some preliminary results

To illustrate how CIM-based computing architecture advances the state-of-the-art, the performance of CIM-based and multicore-based architectures are estimated. We assume a program with a number of parallel instructions executed on the two architectures. Assumptions on multicore architecture and CIM -based architecture are similar to those in [START_REF] Hamdioui | Memristor based computation-in-memory architec ture for data-intensive applications[END_REF]; the multicore architecture consists of 4 cores (ALU only), 32KB cache and 1 GB DRAM; ClM -based architecture consists of one core (ALU only), ClM die with a special computing unit and the data capacity equal to 32KB cache, and 1 GB DRAM. The computations are performed by ALU and the special computing unit. The memory operations are modeled by a memory model based on cache miss rate and DRAM access time similarly as in [START_REF] Hamdioui | Memristor based computation-in-memory architec ture for data-intensive applications[END_REF]. Two metrics are used for the evaluation: (1) Energy Delay Product Efficiency (EDP) defined as the number of executed instruction divided by the energy delay product, and (2) Energy Efficiency (EE) defined as the number of executed instruction divided by consumed energy.

Fig. 6 shows that CIM-based architecture outperforms multicore architecture; the Energy-Delay product efficiency is two orders of magnitude better while Energy efficiency is � 5X better. The improvements are results of significant memory access reduction and the usage of non-volatile memory. The reduction of CPU memory accesses leads to a lower latency and lower energy consumption, while the non volatile memory reduces the static power to practically zero.

  The difTerent types of resistive devices according to difTerent classifications are illustrated in Fig. 1. In Fig. lao the materials suitable for fabricating a resistive element are underlined, in Fig. 1 b the types of conductive path formation are illustrated while in Fig. lc the switching modes are sketched.

  -compliant materials -adaptation after S.Yu, et al. "Metal oxide resistive switching memory," 2011 [41] b) Resistive element classification according to the conductive path: (i) pristine sample, (ii)filamentary path, (iii) homogenous path c) Resistive element classification according to the switching modes: (i) unipolar switching, (ii) bipolar switching

Figure 2 .

 2 Figure 2. Synthesis flow Response time of the circuit defined by the following equation: RespTime = Te * Ne (3)

•

  Nupl and NdOlvnl are the number of memristors in the

,

  -o"'-t-----L +-y----L -t+lfL-t--'-" '+_-' ' 'A B I---"'-+-"-' +--"-+----" +t-.r"-'-!-+-''+_-If 1 = ROFF ;:t 0 = RON J<igure 3. Reset stage

Figure 4 .

 4 Figure 4. Proposed computing architecture IV. MEMRISTTVE DEVICE BASED COMPUTING: MYTH OR REALITY?

Figure 5 .

 5 Figure 5. Example program B. Potential applications

Figure 6 .

 6 Figure 6. Performance comparison between CIM and multicore

TABLE 1 .

 1 EXPERIMENTAL RESULTSExp. INM","plc Cm��hm Sim'i'inl<' OY"rffi,�d ./I.-1inl"m1�

	X0l'5 nJ53 squaTS	S 5 S	22 58 71	15(, 414 515 139	Area R94 4790 (;54(; 1<04	Tinl<' 42 56 56 35	Xb�n (, 8 , 274 (i!) 22R 5(i '" '" '" 215 56 183 40 5 71 19 62 13	�	::;1:�
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Resumo

INSERÇÃO DE DINÂMICA DE GRUPO EM ESCOLAS DE ENSINO FUNDAMENTAL

Keywords:

This article presents a series of reflection and analysis of the use of group dynamics and its importance in the use of this methodology that emphasizes time and space, where they now have another connotation. The application of this method has shown us that many changes occur in frightening pace. Changes in thinking, feeling and acting. The group dynamic is establishing a research field concerned with the study of the nature of the group, to the laws governing its development and for individual-group relations, group-group and group-institutions. The pioneering studies of Kurt Lewin and his disciples, rises field, practically marked the emergence of group dynamics. Learning to work together is a natural tendency of the new globalized world; marked by intolerance and differences. We must learn to live with them. This is definitely a milestone for the use and importance of these techniques.

INTRODUÇÃO

A expressão dinâmica de grupo apareceu pela primeira vez no contexto científico em 1944, num artigo de Kurt Lewin, quando este publicou um estudo consagrado às relações entre a teoria e a prática na Psicologia Social. A dinâmica de grupo começou a se popularizar a partir da segunda guerra mundial, mas foi no final da década de 30 que suas pesquisas se iniciaram. Numa perspectiva histórica, a dinâmica de grupo surgiu como a convergência de determinadas tendências nas ciências sociais, quando se começou a investir no estudo de novas tecnologias para a solução de problemas sociais, que tentavam codificar processos e definir princípios gerais para lidar com pequenos grupos.

O objetivo do estudo da dinâmica de grupo, que é um ramo da psicologia social, consiste em estudar a natureza (ou estrutura) dos pequenos grupos; a dinâmica da vida grupal e o seu funcionamento, assim como o seu processo de desenvolvimento, fenômenos e princípios que lhe regem, as forças psicológicas e sociais que lhe influenciam (como por exemplo, forças de atração, rejeição, coesão, a liderança, a resistência à mudança, a interdependência, etc.).

Segundo o criador do termo "dinâmica de grupo" e seu principal pesquisador, Kurt Lewin (1890-1947), a preocupação central com o estudo dos pequenos grupos em suas dimensões mais concretas e existenciais, é atingir a autenticidade nas suas relações, a criatividade e a funcionalidade nos seus objetivos; para isto, é importante descobrir que estruturas são mais favoráveis, que clima de grupo permite isto, que tipo de liderança é mais eficaz, que técnicas são mais funcionais e facilitadoras, como se dão os mecanismos de atração e rejeição interpessoais, etc. (MAILHIOT, 1991). Sabemos que uma característica do existir humano é reunir-se em grupos sociais. A questão básica da pesquisa em dinâmica de grupo é descobrir como estes grupos se formam e se desenvolvem, como declinam e como influem no comportamento de seus membros.

Somente no século XX foi abandonada uma visão puramente filosófica da compreensão dos pequenos grupos e teve início um estudo experimental. No início deste século predominavam três correntes: a) a individualista, influenciada pela psicanálise freudiana, que enfatiza os determinantes inconscientes no comportamento social; b) a culturalista, enfatizando os determinantes sócio-histórico-culturais dos grupos, defendida pelos antropólogos culturais; c) a behaviorista, que buscava no laboratório o estudo das condutas sociais. Somente a partir de 1936 surgem os trabalhos de Jacob Levy Moreno no ramo da sociometria dos pequenos grupos e no desenvolvimento da psicoterapia de grupo. Assim, as pesquisas de campo de Kurt Lewin, são contemporâneas aos trabalhos de Moreno, acelerando o estudo da dinâmica dos grupos.

O estudo dos fenômenos de grupo, no entanto, é bastante antigo. Encontramos antecedentes desde "A República", de Platão e "A Política", de Aristóteles. Hoje, a dinâmica de grupo abrange estudos multidisciplinares, das áreas de serviço social, psicologia social, psicoterapia de grupo, educação e administração (MAILHIOT, 1991).

As principais teorias que desenvolveram o estudo da dinâmica de grupo, além da teoria de campo, de Kurt Lewin, foram: a) teoria socionômica (que inclui a sociometria, a sociodinâmica e a sociatrianesta última está contida o psicodrama, o sociodrama e a psicoterapia de grupo), de Jacob Levy Moreno; b) teoria da interação, de Bales; c) teoria dos sistemas (de Newcomb, Miller, etc); d) teoria psicanalítica (de Freud, Bion, Eric Berne, etc.); e) teoria cognitiva e construtivista, de Jean Piaget. (CARTWRIGHT & ZANDER, 1975).

Hoje, a dinâmica de grupo é amplamente aplicada no campo da gestão de pessoas, na área de treinamento, desenvolvimento organizacional e de habilidades em relações humanas, desenvolvimento de papéis, etc. É aplicada também em psicoterapias de grupo, em grupos comunitários e em salas de aula. Porém, continua sendo um campo de pesquisa sobre a estrutura, leis e desenvolvimento dos grupos, que se iniciaram com os trabalhos de Moreno e K. Lewin.

A importância do conhecimento e a utilização da psicologia grupal decorrem justamente do fato de que todo indivíduo passa a maior parte do tempo de sua vida convivendo e interagindo com distintos grupos. A dinâmica de grupo é importante porque o ser humano é gregário por natureza e somente existe, ou subsiste, em função de seus interrelacionamentos grupais. Sempre, desde o nascimento, o indivíduo participa de diferentes grupos, numa constante dialética entre a busca de sua identidade individual e a necessidade de uma identidade grupal e social (ZIMERMAN & OSORIO, 1997).

Desde suas mais remotas origens o homem agrupou-se não só para defender-se dos perigos naturais, mas para instrumentalizar seu domínio e poder sobre grupos rivais. O elemento integrador das primeiras experiências grupais foi, sem dúvida, a solidariedade. Para enfrentar ameaças externas, na luta pela sobrevivência, o grupo primordial desenvolveu sentimentos de lealdade e mecanismos de mútua proteção. Se a rivalidade existia na relação entre famílias e tribos distintas, no seio de um mesmo agrupamento humano predominava o elemento solidariedade(SILVA,2016), enquanto persistisse a ameaça externa. Com ou sem estas ameaças, restavam a competição e a luta pelo poder entre os membros de um mesmo grupo (OSORIO, 2000).

DESENVOLVIMENTO

Diante da importância e necessidade do uso das dinâmicas de grupo, o artigo "Inserção de dinâmicas de grupo em escolas de ensino fundamental", tem o objetivo de analisar a utilização de dinâmicas envolvendo um grupo de alunos, seja pequeno ou grande; o primeiro aspecto a ser observado são as técnicas coletivas. Isto quer dizer que elas deverão trazer vantagens diferentes das técnicas usadas para aprendizagem individual e colaborar para outras que não forem capazes de obter apenas individualmente.

Na dinâmica de grupo teremos a capacidade de estudar problemas em equipe, desenvolvendo a colaboração e ouvindo as contribuições dos outros, debatendo e discutindo os vários aspectos do tema, relacionando-os com seus conhecimentos e suas experiências, ampliando seu universo intelectual de tal forma que ao término do trabalho, cada participante possa ter avançado e aprendido mais com relação ao tema em pauta do que tivesse estudando sozinho.

A escolha da metodologia deve ser clara e objetiva, que todos tenham noção do assunto em palta. A dinâmica de grupo constitui um valioso instrumento educacional que pode ser utilizado para trabalhar o ensino-aprendizagem quando se opta por uma concepção de educação que valorize não somente a teoria como a prática.

A experiência do trabalho com dinâmicas promove o encontro de pessoas onde o saber é constituído junto, em grupo. Esse conhecimento deixa de ser coletivizado. Tendo ainda a qualidade de ser um saber onde o indivíduo está envolvido integralmente (afetivamente e intelectualmente) em uma atividade, aonde é constantemente desafiado a analisar criticamente a si e os outros, construindo coletivamente um saber e tentar aplicar seus resultados. "É importante ressaltar que faz parte desse processo a garantia da participação constante de todos os participantes. Só assim todos se sentirão donos do saber alcançado". (PERPÉTUO Gonçalves; 2000.p.2.)

A intenção aqui é elucidar a importância que temos em inserir as dinâmicas de grupo no cotidiano das pessoas, não somente em sala de aula, como em empresas. Então através desse artigo torna-se possível despertar o interesse de professores e gestores no uso das dinâmicas de grupo.

A metodologia está baseada de forma qualitativa, tomando como base em diversos teóricos como: Antunes, Porandoba e outros teóricos, por meio de observações e busca de novas competências à necessidade premente.

A referente pesquisa centra-se na seguinte problemática: o desprestígio do uso das dinâmicas e das pesquisas nessa área, pouco se tem feito talvez pela aversão a pesquisa ou talvez pela tradição e cultura sociopolítica do nosso povo.

Este tema pouco abordado e usado, mas que tem uma grande relevância na integração do indivíduo e o meio em que ele está inserido. A dinâmica de grupo constitui um campo de pesquisa voltado para o estudo da natureza do grupo, para as leis que regem seu desenvolvimento e para relações indivíduo-grupo.

Os estudos de Kurt Lewin e de seus discípulos, sobre campo, praticamente marcaram o aparecimento da dinâmica de grupo. A partir daí, especialistas em dinâmicas de grupo veem empregando grande variedade de métodos de pesquisas. Fazendo com que a vida em grupo fosse trazida para o laboratório e submetida á experimentação controlada.

Este tema foi escolhido tendo em vista à evolução de novos métodos e a facilidade que o emprego das dinâmicas de grupo tem na socialização do indivíduo. A dinâmica de grupo fortalece nossa convicção em relação à importância de seu estudo também como forma privilegiada de submergir o grupo em seu meio. É fundamental o trabalho com dinâmicas, na sua função reveladora e democrática, no direito que o ser tem de revelar-se de uma forma coletiva.

Apresentada por Newcamb, Miller, Stogdill, acentua que o grupo seja um sistema de interação, de comunicação, de encadeamento de posições e de papéis, e principalmente de vários tipos de entrada (in put) e de saída (out put) do sistema.

A dinâmica está vinculada à sociedade em que se origina, assim como todo tipo de arte, pois o artista não consegue ser indiferente da sua realidade. O resultado das dinâmicas de grupo trata claramente a relação entre o meio e o indivíduo, pois o mesmo transmite seus sentimentos e ideias do mundo, levando-o à refletir e até mesmo à mudanças de posição perante a realidade, assim ela tende a auxiliar no processo de transformação social.

A expressão dinâmica de grupo tem servido a uma série de interpretações como:

a ideologias políticas, que acentua a importância da liberação democrática; a participação dos membros de um grupo nas decisões e na solução de problemas. A expressão lança a ideia de coparticipante, como princípio fundamental numa sociedade democrática.

-Também tem sido usada, quando se englobam a um conjunto de técnicas, tais como desempenho de papéis, discussões e interação. "É, pois, por esse conceito, que dinâmica de grupo visa ao estudo e prática de técnicas de trabalho em grupo". Para isso é que os grupos devem desenvolver técnicas e métodos em programa de treinamento. (MINICUCCI, 2002. P. 19) Os estudos de Dewey, o filósofo da democracia, assinalaram, como objetivo da educação, o preparo e a orientação da criança para a vida em sociedade, dessa forma, a escola deixaria de ser simplesmente uma transmissora de conhecimento.

Os trabalhos de Piaget no desenvolvimento do pensamento operativo contribuíram para a utilização do grupo como meio de integração da criança ao meio.

No Brasil, as pesquisas de Lauro de Oliveira Lima desenvolveram a utilização do grupo dentro da linha piagetina. Os educadores vêm cada vez mais utilizando as Dinâmicas de Grupo em suas aulas, dando um sentido novo a elas e ofertando a seus alunos um diferencial.

A dinâmica de grupo empregada em sala de aula tem como finalidade, transformar conteúdos chatos, metódicos, formal em conteúdos apreciáveis, informais e de fácil compreensão. Notase ainda que em dias onde o professor estará aplicando as dinâmicas temos um maior numero de frequência, os alunos mostram-se mais participativos, a interação entre eles é visível.

Parte principal do artigo, pois contém a exposição ordenada do assunto. Divide-se em seções e subseções. O autor deve descrever, abordar e argumentar. É elaborado na forma de revisão de literatura com debate dos autores pesquisados e citados. Pode-se inserir ilustrações (mapas, fotos, gráficos, etc) desde que retratem a situação estudada (GONÇALVES, 2004).

Tamanho da fonte 12, espaço 1,5 entrelinhas. As normas para citações são as mesmas apresentadas na introdução.

Subseção

Entenda como "Dinâmica" uma atividade que possibilita diversas leituras e aplicações e coloca as pessoas em "movimento" de produção, aprendizagem e entretenimento (ao mesmo tempo).

Silvio Luzardo, editor deste Blog, professor de oratória e psicopedagogo (Senac, Santa Catarina, Brasil), aplica diariamente situações dinâmicas no seu método didático e pedagógico, apropriadas e adaptadas ao tema que ministra.

A dinâmica pode desenvolvida em qualquer ambiente e situação em que se reúnam pessoas (jovens, adultos, colaboradores da empresa, familiares, grupos de amigos, sala de aula, treinamento, capacitação, etc). O tema e o tipo de dinâmica devem ser apropriados para a situação que se pretende realizar. Não se deve utilizar uma "dinâmica" sem sentido, só para "animar". Tire sempre um proveito pedagógico dela.

Dinâmica: Escravos de Jó

Objetivo prático: Fazer com que pessoas cantem a canção Escravos de Jó "ao mesmo" tempo em que passam para o seu companheiro da direita uma peça (dominó ou caixa de fósforo são as melhores pois o "ruído" é importante para o ritmo, mas pode ser usado outro objeto maior, como na foto).

Objetivos da Dinâmica Específicos:

-Aprimorar e discutir relacionamentos através da comunicação -Refletir sobre comportamentos e atitudes das pessoas -Descobrir quem são os líderes, quem gosta de mandar, incentivar, inovar; quem é passivo ou apático, etc -Exercício de criatividade e de harmonia -Atividade sugerida para processos de qualidade -Analisar o poder de adaptação, reação e busca de soluções do grupo e dos participantes -Discutir sistemas de atendimento e relacionamento corporativo dentro de uma organização Número de pessoas: ideal entre 8 e 16 (pode ser até o número máximo das pedras do dominó, que são 28) Material: uma peça de dominó ou caixa de fósforo para cada participante

Escravos de Jó tocavam cachangá tira bota deixa do Zé Pereira ficar Guerreiros com guerreiros fazem zig zig zag Guerreiros com guerreiros fazem zig zig zag

CONCLUSÃO

O professor contemporâneo deve ter consigo, sempre à mão, dinâmicas que possibilitem sejam inseridas num contexto apropriado. As dinâmicas são instrumentos pedagógicos de grande valia, principalmente porque integram e interagem as pessoas (sejam jovens ou adultos). Quando aplicadas no momento certo, são ferramentas cognitivas que trabalham o indivíduo como uma unidade própria e como ele se integra no grupo. Na escola, é preciso aplicar dinâmicas de grupo de acordo com a faixa etária, assim pode-se obter um melhor êxito. A dinâmica pode ser encaixada em qualquer conteúdo da grade curricular. Só depende da criatividade do professor ou da professora. Vale a criatividade.

As dinâmicas exercem importância no desenvolvimento físico e mental da criança, o que podem ser observado durante as dinâmicas e brincadeiras, que são prazerosas e despertam interesse além de motivarem o indivíduo de qualquer idade. Assim, tornam-se um importante recurso para a educação, no sentido de desenvolver uma aprendizagem significativa em relação aos conhecimentos do mundo como um todo, bem como sobre as regras de convivência, desenvolvendo valores importantes para a criança e sua vida em sociedade. E, também, pela atividade de despertar a curiosidade do aluno por novos conhecimentos.

Esse artigo tem como finalidade demonstrar a execução da prática de conteúdos estudados no decorrer do curso podendo dessa forma mostrar como esta assimilação bem como a sua aplicabilidade no dia-dia. Dessa forma conclui-se que as funções e metodologias utilizados na elaboração do referido artigo foi de fundamental importância para a conclusão do curso, pois através do mesmo foi possível de forma mais prática e independente a busca por conhecimentos teóricos referentes à de princípios de aplicação de Dinâmicas de grupo em instituições, fazendo valer, na prática, todo o conhecimento teórico, resultando a confirmação de aprimoramento, que possibilitou todas as condições para que assim fosse realizado o plano que resultaria no projeto final.

REFERÊNCIAS