Hui Peng
email: davidpeng1749@gmail.com

Curved, Expanded and Inclined Interference Patterns ---Dynamic Double Slit/Cross-Double Slit Experiments

Keywords: double slit experiments, cross-double slit experiments, interference pattern, light bending, curved interference pattern, expanded interference pattern, inclined interference pattern Content 8. Experiments: Disc-ring 8.1. Apparatus: Disc-Ring 8.2. Expanded Pattern of Rotating Disc-ring 8.3. Disc-ring vs Avago Disc vs Circular Aperture 8.4. Disc-Pattern/Ring-Patterns vs Avago Spot vs Airy Disc/Airy Patterns 9. Discussion 9.1. Static Double Slit Experiments vs Dynamic Double Slit Experiments

The significance of Young's double slit experiments was emphasized as "the only mystery [of quantum mechanics]" (Feynman). In this article, we show three novel universal phenomena of the double slit, cross-double slit and grating experiments: (1) the interference patterns can curve; (2) the interference patterns can expand; and (3) the interference patterns created by the tilt double slits can incline to the axis that is perpendicular to the axis the diaphragm rotates around. The directions of the patterns curved towards are determined by how the diaphragm rotates, i.e., clockwise or counterclockwise. To determine the directions, Right-hand rule and Left-hand rule were proposed. The magnitudes of three phenomena depend on two angles: the original angle of the double slit/crossdouble slit/grating on the diaphragm relative to the rotation axis and the angle the diaphragm rotates. We derived the formular for the expansion of the patterns, and the formular for the inclination angle of patterns. The behaviors of photons are determined by: (1) the orientation of the double slit; (2) which axis the diaphragm rotating around; (3) the direction of rotation, i.e., clockwise or counterclockwise; (4) the angle of the rotation. The Feynman's mystery of the double slit is more mysterious.

Introduction

Young's double slit experiment [START_REF] Ananthaswamy | Through Two Doors at Once[END_REF] [START_REF] Robinson | The Last Man Who Knew Everything[END_REF] contains "the only mystery [of quantum mechanics]" (Feynman) [START_REF] Feynman | The Feynman Lectures on Physics[END_REF]. The nature of photons truly puzzled Einstein. He wrote to M. Besso: "All these 50 years of conscious brooding have brought me no nearer to the answer to the question: What are light quanta?" [START_REF] Rashkovskiy | Is a rational explanation of wave-particle duality possible?[END_REF]. The study on light has been continuously in progress. In 1979, the novel phenomenon of curved Airy beam was predicted theoretically based on Schrödinger's equation [START_REF] Berry | Non-Spreading Wave Packets[END_REF]. In 2007, the experiments confirmed the phenomena [START_REF] Siviloglou | Accelerating finite energy Airy beams[END_REF] [START_REF] Siviloglou | Observation of accelerating Airy beams[END_REF], in which Airy beam transversely accelerates during propagations without external cause. In other word, Airy beam is curved/bended, or it is self-accelerating. The phenomena are also interpreted by Maxwell's equation [START_REF] Kaminer | Nondiffracting Accelerating Wave Packets of Maxwell's Equations[END_REF], and experiments supported the theory [START_REF] Courvoisier | Sending femtosecond pulses in circles: highly nonparaxial accelerating beams[END_REF]. There are review articles, e.g., [START_REF] Cartwright | Light Bends by Itself[END_REF] and [START_REF] Efremidis | Airy beams and accelerating waves: an overview of recent advances[END_REF], to name two. The phenomena of curved Airy beam have attracted considerable attentions not only because its significance in fundamental and experimental optics, but also in varieties of applications.

The light beams are bended mainly in three phenomena, (A) light refraction, (B) light bended by gravity [START_REF] Weinberg | Gravitation and cosmology: principles and applications of the general theory of Relativity[END_REF], (C) Airy beam curves.

In this article, we show the novel phenomena of the curved-light. Namely, the interference patterns of the double slit, cross-double slit and grating experiments are curved transversely, i.e., the fringes distribute along curves, which is significantly different from the above three phenomena of light bending. The curved interference patterns are caused by the different orientations of the diaphragms of the double slit, cross-double slit and grating [START_REF] Peng | Curved Interference Pattern[END_REF] Furthermore, the new phenomena of the expanded and inclined interference patterns were observed, the first time, which indicates that the characteristics of the interference patterns depend on the orientations of the diaphragms of the double slit, cross-double slit [START_REF] Peng | Novel Double Slit and Cross-Double Slit Experiments---Interference Patterns Depending on Orientation of Diaphragm[END_REF] and grating. Table 1 shows the comparison between the static interference pattern of the normal double slit experiments and the dynamic interference pattern of the novel double slit and cross-double slit experiments. Here we referred the nochange interference pattern as the static pattern, while the continuously curve, expand and incline pattern as the dynamic pattern.

The novel phenomena of the curve, expand and incline interference patterns provide the comprehensive data/information to optics and to study the mystery of double slit experiments.

Apparatuses

The experimental apparatus contains a laser source, a protractor and the diaphragms of the cross-double slit (Figure 1) [START_REF] Peng | Cross-Double-Slit Experiment and Extended-Mach-Zehnder Interferometer[END_REF] [16] [START_REF] Peng | Double Slit to Cross-Double Slit to Comprehensive Double Slit Experiments[END_REF], double slit and grating.

Figure 1 Diaphragm of Cross-Double Slit

Experiments of Double Slit

Let us start with a schematic drawing (Figure 2). The double-slit-AB is in the y-z plane, its normal vector is along the x-axis. Slit A and slit B are along the z-direction. Photons travel towards the negative x-direction.

Figure 2 Standard diaphragm of double slit and its pattern

We are interested in whether and how the interference patterns vary with different orientations of the diaphragm of double slit/cross-double slit.

Novel Phenomena: Curved Interference Patterns

The novel phenomenon of the curved interference patterns is attribute to the rotations of the diaphragms around Y-axis. An un-emphasized fact is that the interference patterns of the double slit are along straight line. On the contrary, when the diaphragm in Figure 2 rotates around Y-axis, the double slit creates the curved interference patterns as shown below. The original orientation corresponds to 0 0 rotation (Figure 3a). Rotating the double slit-AB clockwise around Y-axis with different discrete angles, for example, 60 0 (Figure 3b) and 75 0 (Figure 3c). Next step, back to the original orientation and rotating the double slit counterclockwise with different discrete angles (Figure 4), for example, 45 0 (Figure 4a), 60 0 (Figure 4b) and 75 0 (Figure 4c). Note that the degree of bending is a function of "m", where "m" is the m th -order of the fringe, the bigger "m", the larger transvers distance of the pattern curved [START_REF]Personal communication with Dr[END_REF].

Conclusion

Novel Phenomena: Expanded Interference Patterns

The standard optical equation describing the interference pattern of the double slit experiments relates the positions 𝑦 !"#$% of fringes, wave length 𝜆, the spacing 𝑑 between two slits and distance 𝐿 from the double slit to detector/screen as 𝑦 !"#$% = 𝑚 & ' 𝐿. This equation was derived for the situation that the light beam is perpendicular to the plane of the double slit, denoted it as the original orientation.

Let us start with a schematic drawing (Figure 2). The coordinates of slit-A and slit-B are 𝑦 ( = 𝑑/2 and 𝑦 ) = -𝑑/2, the spacing between slits A and B is "d".

We are interested in how the interference patterns vary with different orientations of the diaphragm of double slit. In this section, we study the orientation dependence of the interference patterns of the double-slit, when the diaphragm rotates around Z-axis.

The interference pattern of the double slit at the original position, ∠𝑎 = 0 * , is shown in Figure 5. The diaphragm rotates ∠𝑎 = 60 * from the original orientation (Figure 6).

Figure 6 The diaphragm rotates 60 0 Figure 7 The diaphragm rotates 75 0

The diaphragm rotates ∠𝑎 = 75 * from the original orientation (Figure 7).

Original orientation

Observation: the distances between two fringes become longer.

Conclusion:

(1) Figure 6 and figure 7 show the evolution of the interference patterns varying with orientations of the diaphragm rotating around the z-axis: the larger the rotation angle, the larger the expansion; (2) rotating the diaphragm clockwise and counterclockwise show the same interference patterns. We show, the first time, that the spacings between fringes have expanded attributing to the rotation of the diaphragm. To study, we setup the diaphragm of the tilt double slit in two original orientations:

First original orientation: the diaphragm is in the Y-Z-plane, X-axis is along the normal vector of the plane of the diaphragm, the source is on X-axis; the angle between the double slit and Y-axis is 45 0 clockwise (Figure 8). Second original orientation: the angle between the double slit and Y-axis is 45 0 counterclockwise (Figure 9).

Experiments with First Original Orientation

With the setup of First original orientation (Figure 8), we perform the experiments in two steps.

First step: rotating the diaphragm 60 0 and 75 0 around Y-axis counterclockwise (Figure 10). Observation (Figure 11b and11c): the large the rotation angle of the diaphragm, the larger the expansion of the distance between the fringes, the larger the curvature, and inclined more towards the horizontal axis. The interference pattern curved downwards.

Experiments with Second Original Orientation

With the setup of Second original orientation, we do experiments in two steps.

First step: rotating the diaphragm 60 0 and 75 0 around Y-axis counterclockwise (Figure 12). Second step: rotating the diaphragm 60 0 and 75 0 respectively around Y-axis clockwise (Figure 13). Discussion: in the regular double slit experiment, photons only need to know one factor to behave accordingly:

whether there is a double slit or not. Now we show that photons need to know more factors: (1) the orientation of the double slit on the diaphragm; (2) which axis the diaphragm rotating around; (3) the direction of rotation, i.e., clockwise or counterclockwise; (4) the angle of the rotation.

To determine the directions the interference patters curved towards, Right-hand rule and Left-hand rule have been proposed [START_REF] Peng | Novel Double Slit Experiments: Right-hand Rule and Left-hand Rule ---Interference Patterns Curved, Expanded and Inclined Simultaneously[END_REF]. A mathematical formular was derived to describe the inclination [START_REF] Peng | Inclined, Expanded and Curved Interference Pattern of Double Slit Experiment ---Mathematical Description (1)[END_REF].

Experiments of Cross-Double Slit

Curved and Expanded Interference Patterns: Taking Place Simultaneously

Let us study the orientation-dependence of the interference patterns of the cross-double slit rotating around Zaxis (Figure 14). The apparatus consists of source, cross-double-slit, protractor and screen.

Utilizing a simple cross-double slit that consists of double-slit-AB and double-slit-CD, and double-slit-AB is perpendicular to double-slit-CD (Figure 14).

Curved, Expanded and Inclined Interference Patterns: Taking Place Simultaneously

To study the phenomena that the interference patterns are curved, expanded and inclined simultaneously in the same experiment, we use the apparatus of three double slits crossing at the same spot, one is a vertical double slit, one is horizontal, and one is tilted.

Experiments of Grating

In the grating experiments we utilized the grating of 80 slits/cm.

Curved Interference Patterns

Let us rotate the grating around Y-axis (Figure 20a) clockwise, and then, counterclockwise. We observed the different curved diffraction patterns. Observation: Figure 20c shows the diffraction pattern of the diaphragm rotates 0 0 . Figure 20b shows the diffraction pattern created by the clockwise rotation of the grating. Figure 20d shows the diffraction pattern created by the counterclockwise rotation of the grating.

Expanded Interference Pattern

Figure 21a shows the diffraction pattern before the diaphragm rotating. Now rotating the grating around Z-axis 60 0 (Figure 21b).

Curved, Inclined and Expanded Interference Pattern

To observe the diffraction patterns that are curved, expanded and inclined simultaneously in the same experiment, we rotate the grating around A-axis that is at an angle from Y-axis (Figure 22a). Attribute to the rotations of the diaphragm, either clockwise or counterclockwise, the interference patterns curved.

We need to determine the directions the patterns curved towards first.

Left-hand Rule and Right-hand Rule

The experiments above show complex interference patterns that show the curved interference patterns. To determine the direction of the interference patterns curved towards, the Left-hand Rule and Right-hand Rule were proposed.

Left-hand Rule:

For the curved pattern created by diaphragm rotating counterclockwise.

To determine the direction of the patterns curved towards, point the left thumb to the source, the index finger is aligned with the direction of the original pattern, and the middle finger will point in the direction of the patterns curved towards, which is attribute to the counterclockwise rotation of the diaphragm (Figure 23). To determine the direction of the patterns curved towards, point the right thumb in the direction of the source, the index finger is aligned with the direction of the original pattern, and the middle finger will point in the direction of the patterns curved towards, which is attribute to the clockwise rotation of the diaphragm (Figure 24). To determine the direction of the curved interference patterns, Left-hand Rule and Right-hand Rule are proposed. The experiments in this article obey the rules. However, the underlying physics of the Rules is unclear.

Formular for Expanded Interference Patterns

Derivation of Formular

Let us start with Figure 25. The double-slit-AB is in the y-z plane, slit A and slit B are along the z-direction, its normal vector is along the x-axis and points to source, the coordinates of slit-A and slit-B are 𝑦 ( = 𝑑/2 and 𝑦 ) = -𝑑/2, the spacing between slits A and B is "d".

Figure 25 Setup of double slit experiment

The schematic drawing is the following (Figure 26). The path difference between two waves passing through two slits respectively is (𝑥 + + 𝑥 , ). The requirement of the interference of two waves is that the path difference satisfies the following relation,

𝑥 + + 𝑥 , = 𝑚𝜆, (1)
where,

𝑥 + ≈ 𝑑𝑠𝑖𝑛∠𝑏, (2)
𝑥 , ≈ 𝑑𝑠𝑖𝑛∠𝑎, (3) 𝑠𝑖𝑛∠𝑎
= -! . " = -! /. ! ! 0-! ! = -! . ! + /+0-! ! /. ! ! , (4)
𝑠𝑖𝑛∠𝑏 ≈ 𝑡𝑎𝑛∠𝑏 ≈ -" . " . (5)
Substituting Eqs. ( 2), ( 3), ( 4) and ( 5) into Eq. ( 1), the equations of the orientation-dependence of the constructive/destructive interference pattern of the double slit are [START_REF] Peng | Novel Double Slit and Cross-Double Slit Experiments---Interference Patterns Depending on Orientation of Diaphragm[END_REF],

𝑦 ≡ 𝑦 + + 𝑦 , = 2& ' 𝐿 , :1 + (𝑡𝑎𝑛∠𝑎) , , (6)
𝑦 = (𝑚 + + , ) & ' 𝐿 , :1 + (𝑡𝑎𝑛∠𝑎) , , (7)
where, (𝑡𝑎𝑛∠𝑎) , = -! ! . ! ! , "y" is the position of a bright/dark fringe from the zeroth-order fringe. Taking into account the measurement accuracy, the experiments support the formular.

Testing Formular

Note that:

(1) the traditional formular, 𝑦 = 2& ' 𝐿, is for a special situation that the light source is located on the normal vector of the diaphragm, but the double slit can be at any orientation on the diaphragm;

(2) the derived formular, 𝑦 = 2& ' 𝐿 , :1 + (𝑡𝑎𝑛∠𝑎) , , is for special situations that the double slit parallels to the rotation axis, but the diaphragm can be at any orientation with an angle ∠𝑎;

(3) the general formular should be for the situations that the double slit can be at any orientation on the diaphragm, and the diaphragm can be at any orientation.

Formular for Inclined Interference Patterns

Derivation of Formula

The phenomena take place when the diaphragms rotate, we expect that the formula relates with two angles: one is the original angle of the double slit relative to the rotation axis, another is the rotation angle of the diaphragm. The The double slit-AB is in the Y-Z plane of the diaphragm. The source is on X-axis. The angle ∠a is the original angle between the double alit-AB and Z-axis. Z'-axis is the projection of Z-axis on the detector. The "P" represents the interference pattern formed on the detector. The "D" represents the projection of the double slit-AB on the detector.

The interference pattern "P" is perpendicular to the projection "D". The angle ∠b is the angle between "P" and Z'axis. The angle ∠a' is the angle between "D" and Z'-axis. At the original orientation of the diaphragm, ∠a' is equal to the ∠a (Figure 27). We have

∠𝑏 = 90 * + ∠𝑎′
To describe mathematically, pick an arbitrary point "m" on the "D", making a line mq perpendicular to Z'-axis and meets Z'-axis at the point "q", the line o'q is the projection of the line o'm on Z'-axis. The angle ∠𝑚𝑜′𝑞 is ∠𝑎′ (Figure 28). When the diaphragm rotates around Y-axis counterclockwise, the point m rotates to the point m' (Figure 29).

The vertical length of the line 𝑚𝑞 keeps the same, i.e., 𝑚𝑞 = 𝑚′𝑞′, but the line 𝑜′𝑞 becomes the line 𝑜′𝑞′ that is shorter then 𝑜′𝑞, 𝑜′𝑞 > 𝑜′𝑞′. The line 𝑜′𝑞 represents the projection of the "D" on Z'-axis. The angle ∠𝑚′𝑜′𝑞′ is the angle between the projection of the "D" and Z'-axis. The ∠𝑚𝑜′𝑞 becomes ∠𝑚′𝑜′𝑞, denoted as ∠𝑎". We have ∠𝑚′𝑜′𝑞′ > ∠𝑚𝑜′𝑞, ∠𝑚𝑜 3 𝑞 = ∠𝑎 3 = ∠𝑎, ∠𝑎" > ∠𝑎′, and

𝑡𝑎𝑛∠𝑚 3 𝑜 3 𝑞 3 = 𝑚′𝑞′ 𝑜′𝑞′ = 𝑚𝑞 𝑜′𝑞′ Or 𝑜 3 𝑞 3 = 24 %5#∠2 # " # 4 # . (8)
The interference pattern is perpendicular to the projection of the double slit. Now the angle between the interference pattern "P" and Z'-axis is

90 * + ∠𝑚′𝑜′𝑞′ > 90 * + ∠𝑎. (9)
The larger the diaphragm's rotation angle, the shorter the line 𝑜 3 𝑞 3 and the larger ∠𝑚′𝑜′𝑞′. Thus, the interference pattern inclines to Z'-axis when the diaphragm rotates.

Next to derive the mathematical formula.

Since 𝑚𝑞 = 𝑚′𝑞′, to find the ∠𝑚′𝑜′𝑞′, we need to find 𝑜′𝑞′. For this aim, let us draw the projections of the line o'm' on the X-Z-plane, denoted as the line o'q". the angle of the "D" rotated counterclockwise is ∠𝑞"𝑜′𝑞′ or ∠𝑒 (Figure 30).

Figure 30 Schematics of projection of double slit on X-Z-plane after rotating

When the diaphragm rotates around Y-axis the angle ∠e, the line o'q becomes o'q", 𝑜′𝑞 = 𝑜′𝑞", and

𝑐𝑜𝑠∠𝑒 = 𝑜′𝑞′ 𝑜′𝑞" = 𝑜′𝑞′ 𝑜′𝑞 𝑜′𝑞 3 = (𝑜 3 𝑞)𝑐𝑜𝑠∠𝑒. (10)
Substituting Eq. ( 8) into Eq. ( 10), we obtain

24 %5#∠2 # " # 4 # = (𝑜 3 𝑞)𝑐𝑜𝑠∠𝑒. (11)
Since 24 " # 4 = 𝑡𝑎𝑛∠𝑎, substituting it into Eq. ( 11) we have

𝑐𝑜𝑠∠𝑒 = %5#∠5 %5#∠2 # " # 4 # . Therefore 𝑡𝑎𝑛∠𝑚 3 𝑜′𝑞 3 = %5#∠5 !"$∠7 . ( 12
)
The angle of the pattern relative to the Z-axis is [START_REF] Michel Gondran | Energy flow lines and the spot of Poisson-Arago[END_REF] 90

* + ∠𝑚 3 𝑜′𝑞 3 = 90 * + act Ltan N %5#∠5 !"$∠7 OP. (13)
The larger the rotation angle ∠𝑒, the larger the angle ∠𝑚 3 𝑜′𝑞 3 , namely the interference pattern inclining to Z'axis. The angle between the "P" and Z'-axis is depending on the original angle ∠𝑎 of the double slit-AB and the rotating angle ∠𝑒 of the diaphragm.

Next let us study the situation of rotating the diaphragm clockwise, the projection of the double slit, i.e., the line o'q" is the line o'q' as shown in Figure 31. So, we obtain the same Eq. 13. The double slit-AB is in Y-Z plane of the diaphragm. The source is on X-axis. The angle ∠a is the original angle between the double alit-AB and Z-axis. Z'-axis is the projection of Z-axis on the screen. The "P" represents the interference pattern formed on the screen. The "D" represents the projection of the double slit-AB on the screen. The "P" is perpendicular to the "D". The angle ∠b is the angle between "P" and Z'-axis. The angle ∠a' is the angle between "D" and Z'-axis. At the original orientation of the diaphragm, ∠𝑎′ is equal to the ∠a Figure 32. We have

∠𝑏 = 90 * -∠𝑎′
To describe mathematically, pick an arbitrary point "m" on the D, making a line mq perpendicular to Z'-axis and meets z'-axis at the point q. The line o'q is the projection of the line o'm on Z'-axis. The angle ∠𝑚𝑜′𝑞 is ∠𝑎′ (Figure 33). When the diaphragm rotates around Y-axis clockwise, the point "m" rotates to the point m' (Figure 34).

The vertical length of the line mq keeps the same, i.e., 𝑚𝑞 = 𝑚′𝑞′, but the line 𝑜′𝑞 becomes 𝑜′𝑞′ that is shorter then 𝑜′𝑞, 𝑜′𝑞 > 𝑜′𝑞′. The angle ∠𝑚′𝑜′𝑞′ is the angle between the "D", i.e., 𝑚′𝑜′, and Z'-axis. The ∠𝑚𝑜′𝑞 becomes ∠𝑚′𝑜′𝑞, denoted as ∠𝑎". We have ∠𝑚′𝑜′𝑞′ > ∠𝑚𝑜′𝑞, ∠𝑚𝑜 3 𝑞 = ∠𝑎 3 = ∠𝑎, ∠𝑎" > ∠𝑎′ and

𝑡𝑎𝑛∠𝑚 3 𝑜′𝑞 3 = 𝑚′𝑞′ 𝑜′𝑞′ = 𝑚𝑞 𝑜′𝑞′ Or 𝑜 3 𝑞 3 = 24 %5#∠2 # " # 4 # . ( 14
)
The interference pattern is perpendicular to the double slit. Now the angle between the "P" and Z'-axis is

90 * -∠𝑚 3 𝑜 3 𝑞 3 < 90 * -∠𝑎. (15)
When the rotating angle of the diaphragm becomes larger, the ∠𝑚′𝑜′𝑞′ becomes larger and approaches to 90 * .

Therefore,

90 * -∠𝑚 3 𝑜 3 𝑞 3 → 0 *
Namely, the interference pattern inclines to the Z'-axis.

To find the mathematical description, we need to find the relation between ∠𝑚′𝑜′𝑞′, ∠𝑎 and the diaphragm's rotation angle. Since 𝑚𝑞 = 𝑚′𝑞′, to find the ∠𝑚′𝑜′𝑞′, we need to find 𝑜′𝑞′. For this aim, we draw Figure 35 that is in the X-Z plane.

Figure 35 Schematics of projection of double slit on X-Z'-plane after rotating ∠𝑒.

The 𝑜′𝑞′′ is the projection of the 𝑜′𝑚′ on X-Z'-plane. The 𝑜′𝑞′ is the projection of 𝑜′𝑞′′ on Z'-axis. The angle of the "D" rotated clockwise is ∠𝑞"𝑜′𝑞′ or ∠𝑒.

When the diaphragm rotates around Y-axis an angle ∠𝑒 clockwise, the line 𝑜′𝑞 becomes 𝑜′𝑞′′, i.e., 𝑜′𝑞 = 𝑜′𝑞", and

𝑐𝑜𝑠∠𝑒 = 𝑜′𝑞′ 𝑜′𝑞" = 𝑜′𝑞′ 𝑜′𝑞 𝑜′𝑞 3 = (𝑜 3 𝑞)𝑐𝑜𝑠∠𝑒. (16)
Substituting Eq. ( 14) into Eq. ( 16), we have

(𝑜 3 𝑞)𝑐𝑜𝑠∠𝑒 = 24 %5#∠2 # " # 4 # . ( 17
)
Since 24 " # 4 = 𝑡𝑎𝑛∠𝑎, Eq. ( 17) gives

𝑐𝑜𝑠∠𝑒 = 𝑡𝑎𝑛∠𝑎 𝑡𝑎𝑛∠𝑚 3 𝑜′𝑞 3 Therefore 𝑡𝑎𝑛∠𝑚 3 𝑜′𝑞 3 = %5#∠5 !"$∠7 . ( 18
)
The angle of the interference pattern relative to the Z-axis is [START_REF] Michel Gondran | Energy flow lines and the spot of Poisson-Arago[END_REF],

90 * -∠𝑚 3 𝑜 3 𝑞′ = 90 * -act Ltan N %5#∠5 !"$∠7 OP. ( 19
)
The larger the rotation angle ∠𝑒, the larger the angle ∠𝑚 3 𝑜′𝑞 3 , namely the interference pattern inclining to Z'axis more. The angle between the interference pattern and Z'-axis is depending on the original angle ∠𝑎 of the double slit-AB and the rotating angle ∠𝑒 of the diaphragm.

Next let us rotate the diaphragm counterclockwise, the "D", i.e., o'q", is shown in Figure 36. the projection of the o'q" is the o'q' that is the same as that in Figure 35. Therefore, we will obtain the same mathematical expression, Eq.

19.

Figure 36 Schematics of the diaphragm rotating counterclockwise

Conclusion:

The directions of the diaphragms rotate, either the clockwise or counterclockwise, have no effect on the inclination of the interference pattern.

Note that the formulars, Eq. 13 and Eq. 19, are for the general situation that the double slit can be at any orientation on the diaphragm, and the diaphragm can be at any orientation.

Testing Formular

Now test the mathematical formulars with existing experiments/observations, in which the original angle between the double slit and Y-axis is ∠𝑎 = 45 * . Then rotating the diaphragms, for example, 60 * and 75 * respectively. There are two configurations.

Configuration-1: shown by Figure 27. We have done the experiments and observed the following.

Rotating the diaphragm around Y-axis counterclockwise, for example: (b) 60 0 ; (c) 75 0 (Figure 39). The angles between the interference patterns and Z-axis are shown by the green lines. By rotating the diaphragm of the double slit around one axis, we observe three phenomena simultaneously, namely, the interference patterns curved, expanded and inclined simultaneously (see attached videos, Appendix-3, -4 and -5). To determine the direction of the interference patterns curved towards, Left-hand Rule and Right-hand Rule were proposed [START_REF] Peng | Novel Double Slit Experiments: Right-hand Rule and Left-hand Rule ---Interference Patterns Curved, Expanded and Inclined Simultaneously[END_REF]. The experiments obey the rules. However, the underlying physics of the Rules is unclear. The mathematical formulars were derived for calculating the inclination angles of the interference patterns [START_REF] Peng | Inclined, Expanded and Curved Interference Pattern of Double Slit Experiment ---Mathematical Description (1)[END_REF], attribute to the rotations of the diaphragms. The inclination angles depend on both the original angle of the double slit relative to the rotation axis and the rotation angles of the diaphragm.

The experimental observations support the formulars.

Experiments: Particle vs Wave

To study wave-particle duality and complementarity principle, let us define the term, "the same experiment", as followings: when there is "only one source" emitting light/photons, regardless of the configurations of the experimental apparatus, the experiment is defined as the same experiment.

To study the double slit experiments, the "Virtual Box" model and two postulations were proposed [21] [22].

Let us divide the model into 3 zones: zone-1 (Z-1) is from the source to the diaphragm that is the left boundary of the Virtual Box; the Virtual Box is zone-2 (Z-2); and zone-3 (Z-3) is from the right boundary of the Virtual Box to the detector. The "Virtual Box" represents the diaphragms of the double slit/cross-double slit/grating and their rightside neighborhood (Figure 41).

Figure 41 Double slit Apparatus with "virtual box"

source "Box" detector

Z-1 Z-3 Z-2
Postulate-1 states that, in zone-1, photons behave as particles.

Postulate-2 states that, in zone-3, photons behave as particles. Postulate-2 predicts that in zone-3, each fringe is formed independently and can be formed partially. Indeed, the experiments support the postulate-1 and postulate-2, but it is for the special situation that the light source is on the normal vector of the diaphragms.

Blocking Curved Interference Patterns

The different orientations of the diaphragms/grating create curved, expanded and inclined interference/diffraction patterns, which corresponding to the generalized Virtual Box.

Let us block the curved patterns first. Figure 43 shows the experimental setup. The blocker is placed between the "Virtual Bos" the Z-3. Figure 44a shows the pattern created by the grating when it is at the original orientation. Figure 44b shows the pattern created by the grating when it rotates counterclockwise. Next, blocking the zeroth-order fringe (Figure 44c), blocking the bottom half of the curved pattern (Figure 44d), blocking the top half of the curved pattern (Figure 44e), blocking the left-half of the bottom-half fringes (Figure 44f).

Observation (Figure 44): when parts of the curved patterns are blocked, the rest parts of the patterns keep no change.

Conclusion:

The fringes of the patterns are created independently and partially. Only particle can behave in such way. Postulate-2 is confirmed experimentally again: in zone-3, photons move along predetermined trajectories to form fringes and thus, behave as particles.

It is a challenge to consistently interpret the experiments.

Blocking Expanded Interference Patterns

Let us block the expanded patterns. The setup is the same as shown in Figure 43, but rotates around the perpendicular axis. We have the consistent conclusion.

Conclusion:

The fringes of the patterns are created independently and partially. Only particle can behave in such way. Postulate-2 is confirmed experimentally: in zone-3, photons move along predetermined trajectories to form fringes and thus, behave as particles. We have the consistent conclusion.

Conclusion:

The fringes of the patterns are created independently and partially, which would be expected only if photons behave as particles in Z-3. Postulate-2 is confirmed experimentally: in zone-3, photons move along predetermined trajectories to form fringes and thus, behave as particles. Some of photons form fringes on blockers, while some of photons are distributed like partial of a wave diffraction pattern on the detector.

Experiments: Disc-ring

The disc-ring apparatus consists of the disc-3 and the ring-2. Both disc-3 and ring-2 are formed on the opaque diaphragm, denoted as the disc-ring apparatus. The diaphragm is denoted as "1" in Figure 49a. Figure 49b shows the enlarged real disc-ring apparatus. Figure 49c shows the experimental setup: the diaphragm of the disc-ring is at its original orientation, i.e., the normal vector of the diaphragm is along X-axis, and its pattern (Figure 50). The light beam is perpendicular to the plane of the diaphragm. Observation: there is the disc-shape interference pattern at the center, denoted by number "1" in Figure 50, referred to it as the disc-pattern. A brighter spot is at the center of the disc-pattern. Refer the ring-shape interference patterns as the ring-patterns, denoted by "2" in Figure 50.

Since the distances between two adjacent ring-patterns are the same, we suggest that the formula to describe the interference patterns of the disc-ring experiments is the same as that of the regular double slit. Namely, at the original orientation, the positions of the ring-patterns are

𝑦 = 𝑚 &. ' . ( 20
)
Where "d" is the diameter of disc-3, "L" is the distance between the diaphragm and the screen, "y" is the position of bright ring-patterns, "m" is integers.

Expanded Pattern of Rotating Disc-ring

Then rotate the diaphragm around Z-axis. Figure 51 shows the patterns of the disc-ring rotating 45 0 . Observation: the expansion along Y-axis is larger than that along Z-axis. There is a bright spot at the center of the disc-pattern in Figures 50 and51.

Conclusion:

The larger the rotation angle of the disc-ring, the larger the expansion of the patterns. The expansion is along Y-axis, while along Z-axis, there is no noticeable expansion.

90 * + ∠𝑚 3 𝑜′𝑞 3 = 90 * + act Ltan N %5#∠5 !"$∠7 OP. (13)
Eq. 13 describes that the inclinations of the patterns depend on both angles: the orientation (∠𝑎) of the double slit/cross-double slit and the angle (∠𝑒) of the diaphragm rotates.

Namely, even with the same diaphragm, we will have the continuously changed phenomena when we simply rotate the diaphragm, i.e., the phenomena are dynamically changed. Therefore, we referred those as "dynamic double slit/cross-double slit experiments".

On Light Bending

Postulate-2 has been confirmed for the regular/static double slit experiment. In Section 7, it is shown that postulate-2 holds for novel/dynamic double slit/grating experiments, namely light beam behaves as particles before arriving at the detector and forming the curved pattern.

Based on the confirmed postulate-2, we re-study the interference/diffraction patterns of the double slit/crossdouble slit/grating experiments (Figure 56). Photons behave as particle before land on the detector at the position of each fringe (Figure 56a). Indeed, after passing the diaphragm/grating, photons propagate as particles and land on the position of each fringe respectively, namely, photons bend along Z-direction to land on a straight line to form different fringes (orange-colored light rays and fringes) (Figure 56a).

In this sense, his phenomenon represents the "light bending", referred it as "straight-line light bending".

Namely the "interference pattern" is the "straight-line light bending".

After passing the rotated diaphragm/grating, photons behaving as particles land on the detector along the curved line (green-colored) to form fringes (Figure 56b). This phenomenon represents the "light bending" as well, referred it as "curved-line light bending", or "curved interference pattern".

In the "curved-line light bending", the photons bend not only in the Z direction, but also in Y-direction. The net bending is the combination of both bending. We suggest an interpretation to explain the phenomenon of the curved pattern. A laser source emits photons as a beam of particles (as proved experimentally when testing postulate-1). Those photons travel into the virtual box.

In the language of "wave theory", the function of the "virtual box" is a process of "forming" the distribution of photons as waves.

In the language of "particle theory", the function of the "Virtual box" is a process of "grouping" photons into different groups/streams (denoted the process as "grouping"), which correspond to different fringes respectively. Photons landing on the same fringe are defined as "in the same group". In practice, a "group" of photons propagates as a stream of particles and arrives at the same fringe continuously. Therefore, different groups/streams corresponding to different fringes are formed inside the virtual box. Where the process completed, either in "wave language" or in "particle language", is defined as the right-side boundary of the virtual box.

When coming out of the "virtual box", photons behave as groups/streams of particles (as proved experimentally when testing postulate-2), and follow the trajectories that lead to the different fringes. Although the trajectory of each photon cannot be determined, the trajectory of each group/stream of photons is determined while they are inside the "virtual box". The trajectories of each group/stream of photons are shown by the evolution of each fringe after photons coming out the "virtual box". However, the trajectories cannot be directly observed at the right-side boundary of the "Virtual box", because existing observation equipment can only register photons, but cannot detect the directions of each group/stream of photons simultaneously. The right-side boundary is a point at which, different groups/streams of photons separate. One can observe the patterns only when different groups/streams of photons separate.

In the static double slit/cross-double slit experiments, photons are "forming" or "grouping" to propagate to the straight-line positions of the fringes, i.e., photons bended along a straight line. In the dynamic double slit/crossdouble slit experiments, photons are "forming" or "grouping" to propagate to the curved-line positions of the fringes, or to the expanded positions of the fringes, or to the inclined positions of the fringes. The mechanism of "forming" or "grouping" is mystery. The Feynman's mystery of the double slit experiments becomes more mysterious. Three novel phenomena may be described by Maxwell's equations [START_REF]Personal communication with Dr[END_REF] or quantum theory. It is challenge to interpretate the experiments of the double slit/cross-double slit when the diaphragm rotating, respectively, around Xaxis, Y-axis and Z-axis clockwise and counterclockwise consistently.

  rule, Left-hand rule, Formular for Expansion of Pattern Formular for Inclination of Pattern

.

  Figure 3a rotating 0 0 Figure 3b rotating 45 0 Figure 3c rotating 60 0 Figure 3d rotating 75 0 Figure 3a rotating 0 0 Figure 3b rotating 45 0 Figure 3c rotating 60 0 Figure 3d rotating 75 0

Figure 3

 3 Figure 3 Curved Interference patterns Observation (Figure 3): the interference patterns curved towards the left side.

.Figure 4

 4 Figure 4 Curved Interference patterns Observation: the interference patterns curved towards the right side.

  : the photons' behaviors are significantly different when the diaphragm rotates: (1) clockwise or counterclockwise; (2) different angles. The larger the rotation angle, the larger curvature of the interference patterns.

Figure 5

 5 Figure 5 Diaphragm rotating 0 degree and its pattern

Figure 4a rotating 45 0 Figure

 0 Figure 4a rotating 45 0 Figure 4b rotating 60 0 Figure 4c rotating 75 0

3. 3 .

 3 Novel Phenomena: Curved, Expanded and Inclined Interference Patterns We show the novel double slit experiments in which the following three phenomena emerged simultaneously in the same experiment: (1) the interference patterns inclined; (2) the interference patterns curved; (3) the interference patterns expanded.

.Figure 8

 8 Figure 8 Diaphragm of First Orientation (a) and Original Interference Pattern-1 (b)

Figure 9

 9 Figure 9 Diaphragm of Second Orientation (a) and Original Interference Pattern-2 (b)

.Figure 10

 10 Figure 10 Rotating diaphragm around Y-axis counterclockwise: (a) 0 0 ; (b) 60 0 ; (c) 75 0

.Figure 11

 11 Figure 11 Rotating diaphragm around Y-axis clockwise: (a) 60 0 ; (b) 75 0

Figure 12

 12 Figure 12 Rotating diaphragm around Y-axis counterclockwise: (a) 0 0 ; (b) 60 0 ; (c) 75 0 Observation (Figure 12b and 12c): the large the rotation angle of the diaphragm, the larger the expansion of the distances between the fringes, curved more, and inclined more towards the horizontal axis. The interference pattern curved downwards.

.Figure 13

 13 Figure 13 Rotating diaphragm around Y-axis clockwise: (a) 60 0 ; (b) 75 0 Observation (Figure 13b and 13c): the large the rotation angle of the diaphragm, the larger the expansion of the distances between the fringes, curved more, and inclined more towards the horizontal axis. The interference pattern curved upwards.

Figure 14

 14 Figure 14 Cross-double-slit rotating around z-axis Each double slit of the cross-double slit creates the interference pattern independently [17]. The pattern of the cross-double slit-ABCD is the combination of two interference patterns created respectively by the double slit-AB and double slit-CD. Figure 15 shows the 2D-interference pattern created by the cross-double slit-ABCD at the original orientation.

Figure 15 Diaphragm at 0 0 Figure 16 Diaphragm rotates 60 0 Figure 17 Diaphragm rotates 75 0 Figure 16

 15016017016 Figure 15 Diaphragm at 0 0 Figure 16 Diaphragm rotates 60 0 Figure 17 Diaphragm rotates 75 0 Figure 16 and 17 show the interference pattern of the diaphragm rotating 60 0 and 75 0 around Z-axis. Observation: (1) The double slit-AB creates the expanded interference patterns; (2) the interference pattern created by double slit-CD is curved. Conclusion: the interference patterns are curved and expanded simultaneously in the same experiment, attributing to the rotation of the diaphragm.

Figure 18 Figure 19

 1819 Figure 18 Diaphragm and its original interference pattern Let us rotate the diaphragm around Z-axis counterclockwise.

.Figure 20

 20 Figure 20 rotating grating clockwise and counterclockwise

Figure 21

 21 Figure 21 Expanded diffraction pattern Observation: Figure 21b shows the expanded pattern attributing to the rotation of the grating.

Figure 22 1 .

 221 Figure 22 Curved, expanded and inclined diffraction pattern Observation: Figure 22b shows the diffraction pattern at the diaphragm's original orientation. Figure 22c shows the curved, expanded and inclined diffraction pattern due to the rotation of the grating. The tilted orange line shows that the pattern is curved. The expansion and inclination of the diffraction pattern are shown from the comparison between Figure 22b and 22c.

Figure 23

 23 Figure 23 Left-hand Rule

Figure 24

 24 Figure 24 Right-hand Rule

Figure 3 :

 3 Figure 3: Rotating the diaphragm clockwise: Obey Right-hand rule.

Figure 4 :

 4 Figure 4: rotating the diaphragm counterclockwise: Obey Left-hand rule.

Figure 10 :

 10 Figure 10: rotating the diaphragm counterclockwise: Obey Left-hand rule.

Figure 11 :

 11 Figure 11: rotating the diaphragm clockwise: Obey Right-hand rule.

Figure 12 :

 12 Figure 12: rotating the diaphragm counterclockwise: Obey Left-hand rule.

Figure 13 :

 13 Figure 13: rotating the diaphragm clockwise: Obey Right-hand rule.

Figure 20 :

 20 Figure 20: rotating the grating clockwise (Figure 20b): obey Right-hand rule; rotating the grating counterclockwise (Figure 20d): Obey Left-hand rule.

Figure 26

 26 Figure 26 Schematic drawing for derivation of orientation-dependence of double slit experiments

Figure 5 :

 5 Figure 5: The diaphragm is at the original position, ∠𝑎 = 0 * , Eq. (6) becomes the normal formular, 𝑦 = 2& ' 𝐿.

Figure 6 : 2 2&'

 62 Figure 6: The diaphragm rotates 60 0 , Eq. (6) gives 𝑦 ≈ 2 2& ' 𝐿 , .

Figure 7 :

 7 Figure 7: The diaphragm rotates 75 0 , Eq. (6) gives 𝑦 ≈ 3.86 2& ' 𝐿 , .

Configuration- 1 :

 1 Figure 27 Schematic of Inclination of Double Slit Experiment-1

Figure 28

 28 Figure 28 Projection of double slit Figure 29 projection of double slit on detector before rotating on detector after rotating counterclockwise

Figure 31 Configuration- 2 :

 312 Figure 31 Schematics of the diaphragm rotating clockwise Conclusion: The directions of the diaphragms rotate, either the clockwise or counterclockwise, have no effect on the inclination of the interference pattern. Configuration-2: Configurataion-2 is shown in Figure 32 below:

  Figure 33 Projection of double slit

Figure 27 configuration- 1 First

 1 Figure 27 configuration-1 First, define how to measure the angle of the curved interference patterns: 1) drawing a line connecting the same-order fringes, for example, the green inclined lines in the following Figures. Then draw a horizontal line representing Z'-axis. The angle between those two lines is the inclination angle of the interference pattern relative to Z'-axis.Rotating the diaphragm around Y-axis counterclockwise: for example, (b) 60 0 ; (c) 75 0 (Figure37). The angles between the interference patterns and Z-axis are shown by green lines.

Figure 37

 37 Figure 37 Rotating diaphragm around Y-axis counterclockwise: (b) 60 0 ; (c) 75 0 Observation: the large the rotation angle of the diaphragm, the interference patterns inclined closer to the horizontal axis. The interference pattern curved upwards, which agree with Left-hand Rule. Rotating the diaphragm around Y-axis clockwise, for example: (b) 60 0 ; (c) 75 0 (Figure 38). The angles between the interference patterns and Z-axis are shown by the green lines.

Figure 38 Configuration- 2 :

 382 Figure 38 Rotating diaphragm around Y-axis clockwise: (b) 60 0 ; (c) 75 0 Observation: the large the rotation angle of the diaphragm, the interference patterns inclined closer to the horizontal axis. The interference pattern curved downwards, which agree with Right-hand Rule. Eq. (13) predicts that the inclined angles of the interference patterns, attribute to the diaphragm rotating 60 0 and 75 0 , are 153 0 and 165 0 , respectively, either clockwise or counterclockwise, ∠𝑏 = 90 * + ∠𝑎 3 = 153 * (For rotating 60 0 ) ∠𝑏 = 90 * + ∠𝑎 3 = 165 * (For rotating 75 * ) Configuration-2: shown by Figure 32.

Figure 32

 32 Figure 32

Figure 39 Figure 40

 3940 Figure 39 Rotating diaphragm around Y-axis counterclockwise: (b) 60 0 ; (c) 75 0

Figure 42

 42 Figure 42 Generalized Models of Virtual Box Figure 42a and 42b show the rotating virtual boxes, where the green-colored circular arrows indicate the directions of the rotations, counterclockwise and clockwise, the orange-colored arrows indicate the rotation axis.

FigureFigure 44

 44 Figure 43 Experimental setup

Figure 45

 45 Figure 45 Original pattern and pattern of rotating 60 0 Figure 45 shows the original pattern of the grating and expanded pattern of grating after rotating 60 0 . Let us block the pattern at different parts.

7. 3 .Figure 47

 347 Figure 47 Original pattern and pattern of rotating 60 0Figure 47a and 47b show the original pattern of the grating and the inclined/expanded pattern of the grating after rotating 60 0 respectively. Let us block the pattern at different parts.

  Figure 49 Disc-Ring apparatus

Figure 50

 50 Figure 50 Disc-ring and disc-pattern: rotating 0 0

Figure 50

 50 Figure 50 Diaphragm rotating 0 0 Figure 51 Diaphragm rotating 45 0 Figure 52 and Figure 53 show the patterns of the disc-ring rotating 60 0 and 75 0 respectively.

Figure 52 Rotating 60 0

 0 Figure 52 Rotating 60 0 Figure 53 Diaphragm Rotating 75 0

.Figure 56

 56 Figure 56 Normal/static vs novel/dynamic grating experiments

Figure 57 "

 57 Figure 57 "Straight-line light bending" vs "Curved-line light bending"

  We show the new phenomena that the interference patterns of the double slit and cross-double slit experiments depend on the orientation of the diaphragms. When the diaphragms rotate: (1) around different axis; (2) clockwise or counterclockwise; (3) different angles, the photons' behaviors are significantly different: curved, expanded and inclined, or two of three phenomena, or all three phenomena, taking place simultaneously in the same experiment. We derived the formulars to describe the expansion and inclination, and proposed Right-hand rule and Lefthand rule to determine the direction of the interference patterns curved towards. The significances of the novel double slit and cross-double slit experiments of this article are: (1) discovered, the first time, three novel phenomena, i.e., the patterns curved, expanded and inclined; and (2) provide comprehensive phenomena/information for developing theoretical model to explore the mystery of the double slit experiments. Let us compare the phenomena of the curved light. theory Right-hand rule, Left-hand rule, Formular for Expansion of Pattern Formular for Inclination of Pattern applications Gravity lenses biomedical imaging, filamentation

Table 1

 1 Normal/Static Double Slit vs Novel/Dynamic Double Slit/Cross-Double Slit Experiments

Acknowledgement

The author acknowledges Mr. Paul Sorich for making all diaphragms of cross-double slits.

Figure 54a shows Avago disc that is the circular opaque disc; Figure 54b is the disc-ring apparatus proposed in this article; Figure 54c shows the circular aperture.

The disc-ring can be considered as the combination of Avago disc and the circular aperture. The disc-pattern and the ring-patterns have the similar shape as that of Airy-disc and Airy-patterns.

Note that, for the disc-ring apparatus, the distances between two adjacent ring-patterns are the same. For circular aperture, however, the distances between two adjacent Airy patterns are different.

Discussion

Static Double Slit Experiments vs Dynamic Double Slit Experiments

To distinguish the regular double slit experiments and the novel double slit experiments and to understand the experiments, the regular double slit experiments are referred as the "static double slit experiments", while the novel double slit experiments are referred as "dynamic double slit experiments". The reasons are the following.

The regular double slit experiments are described by the formular,

Where "d" is the spacing between two slits, and is the only parameter related with the diaphragm. For a given diaphragm, i.e., the "d" is fixed, the experimental result is stationary.

On the contrary, the novel double slit/cross-double slit experiments contain three novel phenomena, curved, expanded and inclined interference patterns.

The expansion of the patterns of the novel double slit/cross-double slit experiments are described by the formular,

For a given diaphragm, beside the "d", the distances between fringes are also depended on both the orientation (∠𝑎) of the diaphragm and the orientation of the double slit on the diaphragm.

The inclinations of the patterns are described by the formular,


Nicolas Marie

PROJECTION ESTIMATORS OF THE STATIONARY DENSITY OF A DIFFERENTIAL EQUATION DRIVEN BY THE FRACTIONAL BROWNIAN MOTION

, an oracle inequality is established for the adaptive estimator.

Introduction

Consider the differential equation (1)

X t = X 0 + t 0 b(X s )ds + σB t ; t ∈ R + ,
where X 0 is a real-valued random variable, B = (B t ) t∈R+ is a fractional Brownian motion of Hurst index H ∈ (0, 1), b : R → R is a continuous map and σ ∈ R * := R\{0}. Throughout the paper, it is assumed that Equation (1) has a unique stationary solution, which means in particular that there exists a unique random variable X 0 such that X t has the same distribution than X 0 for every t ∈ R + . A sufficient condition is given at Section 2.

For over two decades, many authors have investigated statistical questions related to differential equations driven by the fractional Brownian motion (fDE).

A large part of the papers published on statistical inference for fDEs deals with parametric estimators of the drift function b when H > 1/2. In Kleptsyna and Le Breton [START_REF] Kleptsyna | Some Explicit Statistical Results About Elementary Fractional Type Models[END_REF] and Hu and Nualart [START_REF] Hu | Parameter Estimation for Fractional Ornstein-Uhlenbeck Processes[END_REF], continuoustime estimators of the drift parameter in Langevin's equation are studied. Kleptsyna and Le Breton [START_REF] Kleptsyna | Some Explicit Statistical Results About Elementary Fractional Type Models[END_REF] provide a maximum likelihood estimator, where the stochastic integral with respect to the solution to Equation (1) returns to an Itô integral. In [START_REF] Tudor | Statistical Aspects of the Fractional Stochastic Calculus[END_REF], Tudor and Viens extend this estimator to equations with a drift function depending linearly on the unknown parameter. On the maximum likelihood estimator in fDEs with multiplicative noise, see Mishura and Ralchenko [START_REF] Mishura | On Drift Parameter Estimation in Models with Fractional Brownian Motion by Discrete Observation[END_REF]. Hu and Nualart [START_REF] Hu | Parameter Estimation for Fractional Ornstein-Uhlenbeck Processes[END_REF] provide a least squares estimator, where the stochastic integral with respect to the solution of Equation ( 1) is taken in the sense of Skorokhod. In [START_REF] Hu | Drift Parameter Estimation for Nonlinear Stochastic Differential Equations Driven by Fractional Brownian Motion[END_REF], Hu, Nualart and Zhou extend this estimator to equations with a drift function depending linearly on the unknown parameter. Tindel and Neuenkirch [START_REF] Neuenkirch | A Least Square-Type Procedure for Parameter Estimation in Stochastic Differential Equations with Additive Fractional Noise[END_REF] provide a discrete-time least squares type estimator defined by an objective function allowing to make use of the main result of Tudor and Viens [START_REF] Tudor | Variations and Estimators of Self-Similarity Parameters via Malliavin Calculus[END_REF] on the rate of convergence of the quadratic variation of the fractional Brownian motion. In [START_REF] Panloup | A General Drift Estimation Procedure for Stochastic Differential Equations with Additive Fractional Noise[END_REF], Panloup, Tindel and Varvenne extend the results of [START_REF] Neuenkirch | A Least Square-Type Procedure for Parameter Estimation in Stochastic Differential Equations with Additive Fractional Noise[END_REF] under much more flexible conditions. More recently, nonparametric methods were investigated to estimate the drift function b in Equation [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF]. For instance, Saussereau [START_REF] Saussereau | Nonparametric Inference for Fractional Diffusions[END_REF] and Comte and Marie [START_REF] Comte | Nonparametric Estimation in Fractional SDE[END_REF] study the consistency of continuous-time Nadaraya-Watson type estimators of b.

The common point of all the references mentioned above is that the existence and uniqueness of the stationary solution to Equation ( 1) is required. Even if to estimate the distribution of the stationary solution is not necessary to study estimators of b, this is a very important question already investigated via kernel based methods in Bertin et al. [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF]. Precisely, when the stationary solution has a density f with respect to Lebesgue's measure, the authors establish a risk bound on Parzen's estimator of f and provide an oracle inequality for an adaptive estimator obtained via a Goldenshluger-Lepski type method. This is a nice application of a powerful concentration inequality for Lipschitz functionals of discrete samples of X also established by Bertin et al. in [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF].

Let S m be the vector space generated by an orthonormal family B m = {ϕ 1 , . . . , ϕ m } of L 2 (I, dr), where I ⊂ R is an interval, and consider t 0 , . . . , t n > 0 such that t i := i∆ n for every i ∈ {1, . . . , n}, ∆ n > 0,

lim n→∞ ∆ n = 0 and lim n→∞ n∆ n = ∞.
Our paper deals with the following projection estimator of f :

f m = f m,n := m j=1 [ θ m,n ] j ϕ j with θ m,n := 1 n n i=1 ϕ j (X ti ) j∈{1,...,m} .
Precisely, a concentration inequality on the supremum of the empirical process is derived from Bertin et al. [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF], Theorem 1 in Section 2. Then, a risk bound on f m is established in Section 3. Section 4 deals with a model selection method and an oracle inequality for the adaptive estimator. Finally, some basic numerical experiments are provided at Section 5. As in the i.i.d. context, the main advantage of the projection based approach for one-dimensional fDEs is that the optimization problem defining the model selection method is numerically easier to solve than the one defining the Goldenshluger-Lepski method because it involves only one variable. Note also that in the kernel based approach, even in dimension 1 in the i.i.d. context, there is no simple model selection method as (2). This is the main advantage of the adaptive estimator studied in this paper with respect to the kernel-based adaptive estimator of Bertin et al. [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF]. In fact, the same advantage than in the i.i.d. context (see the remarks at the end of Section 4 for details).

Notations:

(1) Throughout the paper, R n is equipped with the distance d 1 defined by

d 1 (x, y) := n i=1 |y i -x i | ; ∀x, y ∈ R n .
(2) For any metric space E, Lip(E; R) is the space of Lipschitz continuous maps from E into R, equipped with its usual semi-norm, always denoted by . Lip for the sake of simplicity. (3) The space C 0 (E; R) is equipped with the uniform norm, always denoted by . ∞ for the sake of simplicity. (4) The space L 2 (I, dr) is equipped with its usual scalar product ., . . The associated norm is denoted by . . (5) Throughout the paper, R I is identified to {ϕ : R → R : supp(ϕ) = I}.

Preliminaries: stationary solutions of the fDE

This section deals with existing results on the existence and uniqueness of the stationary solution to Equation ( 1) under a dissipativity condition on the drift function b, and then with some consequences of a concentration inequality for Lipschitz functionals of (X t1 , . . . , X tn ) due to Bertin et al. [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF]. 1) has a unique stationary solution X = (X t ) t∈R+ (see Hairer [START_REF] Hairer | Ergodicity of Stochastic Differential Equations Driven by Fractional Brownian Motion[END_REF]) and X 0 has a density f with respect to Lebesgue's measure (see Bertin et al. [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF], Proposition 1).

Example 2.2. A famous example of fDE satisfying Assumption 2.1 is the fractional Langevin equation (b = -θId R with θ > 0). Its solution is the fractional Ornstein-Uhlenbeck process and, in this case, the stationary density f is Gaussian (see Cheridito et al. [START_REF] Cheridito | Fractional Ornstein-Uhlenbeck Process[END_REF]).

The following theorem provides a concentration inequality for Lipschitz functionals of (X t1 , . . . , X tn ) (see Bertin et al. [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF], Theorem 1).

Theorem 2.3. Under Assumption 2.1, there exists a constant c 2.3 > 0, not depending on n, such that for every F ∈ Lip(R n ; R) and r > 0,

P(F (X t1 , . . . , X tn ) -E(F (X t1 , . . . , X tn )) > r) exp - r 2 c 2.3 F 2 Lip n a H ∆ -b H n with a H = (2H) ∨ 1 and b H = 1 ∧ (2 -2H).
Now, let us state a consequence of Theorem 2.3 on the (centered) empirical process, already established in Bertin et al. [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF] (see Corollary 1).

Corollary 2.4. Under Assumption 2.1, for every ϕ ∈ Lip(R) and r > 0,

P 1 n n i=1 [ϕ(X ti ) -E(ϕ(X ti ))] > r exp - r 2 c 2.3 ϕ 2 Lip (n∆ n ) b H .
Finally, because Corollary 2.4 is not sufficient to establish a risk bound on the adaptive estimator at Section 4, let us derive a concentration inequality on the supremum of the empirical process from Theorem 2.3 under the following assumption of f . Assumption 2.5. The stationary density f belongs to L 2 (R, dr).

Remark. It is plausible that f fulfills Assumption 2.5 for a wide class of drift functions fulfilling Assumption 2.1, but this problem is out of the scope of the present paper. However, let us provide some examples. On the one hand, since it is Gaussian, f fulfills Assumption 2.5 when (1) is the fractional Langevin equation (b = -θId R with θ > 0). On the other hand, when H = 1/2 and σ > 0, as mentioned in Bertin et al. [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF] (see Remark 1),

f (x) = c U,σ exp U (x) 2σ 2 ; ∀x ∈ R
where U is a primitive function of b and c U,σ is a positive constant. By Assumption 2.1, there exist

c 1 , c 2 > 0 such that U (x) -m b x 2 + c 1 x + c 2 ; ∀x ∈ R.
Then, f fulfills Assumption 2.5.

Corollary 2.6. Consider F ⊂ Lip(R) such that

c F := sup ϕ∈F ϕ Lip < ∞.
Under Assumptions 2.1 and 2.5, for every ϕ ∈ F, consider the empirical process

ν n (ϕ) := 1 n n i=1 [ϕ(X ti ) -m(ϕ)] with m(ϕ) := ϕ, f .
Then, for every

h E sup ϕ∈F |ν n (ϕ)|
and every r > 0,

P sup ϕ∈F |ν n (ϕ)| -h > r exp - r 2 c 2.3 c 2 F (n∆ n ) b H .
Proof. Let F : R n → R be the map defined by

F (x) := sup ϕ∈F 1 n n i=1 [ϕ(x i ) -m(ϕ)] ; ∀x ∈ R n .
For any x, y ∈ R n , by the triangle inequality for the uniform norm,

|F (y) -F (x)| 1 n sup ϕ∈F n i=1 [ϕ(y i ) -m(ϕ)] - n i=1 [ϕ(x i ) -m(ϕ)] 1 n sup ϕ∈F n i=1 |ϕ(y i ) -ϕ(x i )| 1 n d 1 (x, y) sup ϕ∈F ϕ Lip .
Then, F ∈ Lip(R n ; R) and

F Lip c F n .
Therefore, by Theorem 2.3 and since 2 -a H = b H , for every r > 0,

P sup ϕ∈F |ν n (ϕ)| -h > r P(F (X t1 , . . . , X tn ) -E(F (X t1 , . . . , X tn )) > r) exp - r 2 c 2.3 c 2 F (n∆ n ) b H .

Risk bound on the projection estimators

This section deals with a risk bound on f m obtained via the concentration inequality on the empirical process stated in Corollary 2.4.

In the sequel, f m is the orthogonal projection of f on S m (in L 2 (I, dr)), and B m fulfills the following assumption.

E( f m -f 2 ) min t∈Sm t -f 2 + 2c 2.3 mL(m) (n∆ n ) b H with L(m) := m j=1 ϕ j 2 Lip .
Proof. First of all, for any x ∈ I,

E(| f m (x) -f (x)| 2 ) = (E( f m (x)) -f (x)) 2 + var( f m (x)).
Let us express well the bias term, and then give a suitable bound for the variance term. On the one hand,

E( f m (x)) = 1 n m j=1 n i=1 E(ϕ j (X ti ))ϕ j (x) = m j=1 f, ϕ j ϕ j (x) = f m (x).
So,

I (E( f m (x)) -f (x)) 2 dx = f m -f 2 = min t∈Sm t -f 2 .
On the other hand, consider ϕ(x, .) := m j=1 ϕ j (.)ϕ j (x).

By Corollary 2.4 applied to ϕ(x, .),

var( f m (x)) = 2 ∞ 0 rP 1 n n i=1 [ϕ(x, X ti ) -E(ϕ(x, X ti ))] > r dr 4 ∞ 0 r exp - r 2 c 2.3 ϕ(x, .) 2 Lip (n∆ n ) b H dr = 2c 2.3 Γ(1) ϕ(x, .) 2 Lip (n∆ n ) b H .
Moreover, by Jensen's inequality and since B m is an orthonormal family of L 2 (I, dx),

I ϕ(x, .) 2 Lip dx I   m j=1 ϕ j Lip |ϕ j (x)|   2 dx m m j=1 ϕ j 2 Lip = mL(m).
Therefore,

E( f m -f 2 ) min t∈Sm t -f 2 + 2c 2.3 mL(m) (n∆ n ) b H .
Example 3.3. Assume that I = [0, 1] and that B m is the trigonometric basis. Precisely, for every x ∈ I, ϕ 1 (x) = 1, and for every j ∈ N such that 2j + 1 m, ϕ 2j (x) = √ 2 cos(2πjx) and ϕ 2j+1 (x) = √ 2 sin(2πjx). Then, there exists a constant c 3.3 > 0, not depending on m and n, such that

L(m) = m j=1 ϕ j 2 ∞ c 3.3 m 3 .
So, the variance term in the risk bound on f m stated in Proposition 3.2 is of order m 4 (n∆ n ) -b H . Note that the variance term in the risk bound on Parzen's estimator of bandwidth h > 0 obtained in Bertin et al. [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF], Proposition 3, is of same order h -4 (n∆ n ) -b H .

Model selection

As in the classic projection density estimation framework, note that

f m = arg min ϕ∈Sm γ n (ϕ) with γ n (ϕ) := ϕ 2 - 2 n n i=1 ϕ(X ti ) ; ∀ϕ ∈ S m .
So, for the proposal set

M := {m ∈ N * : mL(m) (n∆ n ) b H } with N * = N\{0},
and a hyper-parameter K > 0, it is natural to consider the adaptive estimator f m of f , where

(2) m = arg min m∈M { γ n ( f m ) + pen(m)} with pen(m) := K (m + 1)L(m) (n∆ n ) b H ; ∀m ∈ M.
In order to provide an oracle inequality for f m at Theorem 4.5, let us first establish the following technical lemma. So,

ϕ Lip m j=1 |a j | • ϕ j Lip   m j=1 a 2 j   1/2   m j=1 ϕ j 2 Lip   1/2 = L(m) 1/2 .
Therefore,

c Fm = sup ϕ∈Fm ϕ Lip L(m) 1/2 .
In the sequel, the S m 's are nested:

Assumption 4.2. For every m, m ∈ M, if m m , then S m ⊂ S m .
Moreover, L(.) fulfills the following assumption.

E( f m -f 2 ) c 4.5,1 min m∈M { f m -f 2 + pen(m)} + c 4.5,2 (n∆ n ) b H .
Proof. The proof is dissected in two steps. In the first one, with the same arguments than in the classic projection density estimation framework (see Comte [START_REF] Comte | Nonparametric Estimation[END_REF], Theorem 5.2 or Massart [START_REF] Massart | Concentration Inequalities and Model Selection[END_REF], Chapter 7), it is established that for any m ∈ M,

f m -f 2 3 f m -f 2 + 4pen(m) + R m,n ,
where R m,n is a remainder term. In the second step, it is established that E(R m,n ) is of order (n∆ n ) -b H thanks to Corollary 2.6 and Lemma 4.1.

Step 1. Note that

γ n ( f m ) + pen( m) γ n (f m ) + pen(m) ; ∀m ∈ M and γ n (ϕ) -γ n (ψ) = ϕ -f 2 -ψ -f 2 -2ν n (ϕ -ψ) ; ∀ϕ, ψ ∈ Lip(I; R).
Then, for any m ∈ M, since 2uv u 2 + v 2 for every u, v ∈ R + , and since

S m + S m =    m j=1 θ j ϕ j + m j=1 θ j ϕ j ; θ 1 , . . . , θ m , θ 1 , . . . , θ m ∈ R    ⊂ S m∨m ; ∀m ∈ M by Assumption 4.2, f m -f 2 f m -f 2 + pen(m) + 2 • 1 2 f m -f m • 2ν n f m -f m f m -f m -pen( m) f m -f 2 + pen(m) + 1 4 f m -f m 2 + 4 sup ϕ∈F m∨ m |ν n (ϕ)| 2 -pen( m).
For every m ∈ M, consider

p(m, m) := K 4 • ((m ∨ m) + 1)L(m ∨ m) (n∆ n ) b H 1 4
(pen(m) + pen(m)).

So,

f m -f 2 f m -f 2 + pen(m) + 1 4 f m -f m 2 +4 sup ϕ∈F m∨ m |ν n (ϕ)| 2 -p(m, m) + + 4p(m, m) -pen( m) f m -f 2 + 2pen(m) + 1 2 ( f m -f 2 + f m -f 2 ) +4 m∈M sup ϕ∈Fm∨m |ν n (ϕ)| 2 -p(m, m)
+ and then,

f m -f 2 3 f m -f 2 + 4pen(m) + 8 m∈M sup ϕ∈Fm∨m |ν n (ϕ)| 2 -p(m, m) + .
Step 2. For any m ∈ M, by Corollary 2.4 (as in the proof of Proposition 3.2), and since K 16c 2.3 , Thus, by Corollary 2.6,

E sup ϕ∈Fm∨m |ν n (ϕ)| 2 m∨m j=1 E(ν n (ϕ j ) 2 ) = m∨m j=1 var(ν n (ϕ j )) 2c 2.3 1 (n∆ n ) b H m∨m j=1 ϕ j 2 Lip K 8 • L(m ∨ m) (n∆ n ) b H =: h(m, m). Now, consider p(m, m) := 1 2 p(m, m) -h(m, m) = K 8 • (m ∨ m)L(m ∨ m) (n∆ n ) b H > 0.
E sup ϕ∈Fm∨m |ν n (ϕ)| 2 -p(m, m) + 2 exp - p(m, m)(n∆ n ) b H c 2.3 c 2 Fm∨m ∞ 0 exp - r(n∆ n ) b H c 2.3 c 2 Fm∨m dr = 2Γ(1)c 2.3 1 (n∆ n ) b H c 2 Fm∨m exp - p(m, m)(n∆ n ) b H c 2.3 c 2 Fm∨m .
Moreover, by Assumption 4.3, there exist three constants c 1 , c 2 , c 3 > 0, not depending on m and n, such that

m∈M c 2 Fm∨m exp - p(m, m)(n∆ n ) b H c 2.3 c 2 Fm∨m m∈M L(m ∨ m) exp - K 8c 2.3 (m ∨ m) = mL(m) exp - K 8c 2.3 m + m∈M : m>m L(m) exp - K 16c 2.3 m exp - K 16c 2.3 m c 1 + c 2 m∈M exp - K 16c 2.3 m c 3 .
Therefore,

E( f m -f 2 ) min m∈M {3 f m -f 2 + 4pen(m)} + 2c 2.3 c 3 (n∆ n ) b H .
Remarks:

(1) Note that the penalty and the remainder term in the risk bound in Theorem 4.5 are of same order than in Bertin et al. [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF], Theorem 3. (2) As in the i.i.d. context, since it involves only one variable, the main advantage of the projection based approach for one-dimensional fDEs is that Problem (2) is numerically easier to solve than the optimization problem defining the Goldenshluger-Lepski method. Note also that in the kernel based approach, even in dimension 1 in the i.i.d. context, there is no simple model selection method as [START_REF] Chagny | Warped Bases for Conditional Density Estimation[END_REF]. Recently, in [START_REF] Lacour | Estimator Selection: a New Method with Applications to Kernel Density Estimation[END_REF], Lacour, Massart and Rivoirard have provided a bandwidth selection method, called PCO method, bypassing the mentioned drawbacks of Goldenshluger-Lepski's method, but to extend the PCO method to the fDE framework requires more than Bertin et al. [START_REF] Bertin | Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion[END_REF], Theorem 1. An extension of the concentration inequality for U -statistics of order 2 of Houdré and Reynaud-Bouret [START_REF] Houdré | Exponential Inequalities, with Constants, for U -Statistics of Order Two[END_REF] from the i.i.d. context to (X t1 , . . . , X tn ) is required. (3) Note that without additional arguments, one can extend the result of our paper to multidimensional fDEs for isotropic projection estimators. However, to extend it to anisotropic projection estimators requires a Goldenshluger-Lepski type procedure again (see Chagny [START_REF] Chagny | Warped Bases for Conditional Density Estimation[END_REF] for this type of method).

Basic numerical experiments

In this section, for H ∈ [1/2, 1), the efficiency of our projection estimator of the stationary density f is evaluated when (1) is the fractional Langevin equation (b = -θId R with θ > 0). In this case,

f (x) = 1 2πσ 2 θ -2H HΓ(2H) exp - x 2 2σ 2 θ -2H HΓ(2H)
; ∀x ∈ R.

The fractional Brownian motion is simulated via the Decreusefond-Lavaud method (see Decreusefond and Lavaud [START_REF] Decreusefond | Simulation of the Fractional Brownian Motion and Application to the Fluid Queue[END_REF]) for H = 0.5 and H = 0.7 along the dissection {kT /n ; k = 0, . . . , n} of [0, T ] with T = 100 and n = 10 3 . The fractional Langevin equation is simulated with the initial condition x 0 = 5, θ = 10 and σ = 0.25 by using the step-n Euler scheme X (n) defined by

X (n) 0 = x 0 X (n) k+1 = X (n) k -θX (n) k T /n + σ(B t k+1 -B t k ) ; k = 0, . . . , n -1 .
Since under Assumption 2.1 the solution to Equation (1) with initial condition x 0 converges pathwise and exponentially fast to its stationary solution when t → ∞, for T and n large enough, the error induced by considering datasets generated by the Euler scheme X (n) is not that significant. In each case (H = 0.5 or H = 0.7), for the trigonometric basis, our projection estimator is computed on 10 independent datasets along the dissection {j/N ; j = -N, . . . , N } of [-1, 1] with N = 70. The average MISE is provided.

Case H = 0.5. On Figure 1, the 10 estimations (dashed black curves) of f (red curve) generated by f m are plotted for m = 30, leading to an average MISE of 0.572, lower than for m = 25 or m = 35.

Assumption 2 . 1 .

 21 The function b belongs to C 1 (R), b is bounded and there exists m b > 0 such that b (x) -m b ; ∀x ∈ R. Under Assumption 2.1, Equation (

Assumption 3 . 1 .

 31 The ϕ j 's are bounded and Lipschitz continuous functions. Now, let us establish the main result of this section: a risk bound on f m . Proposition 3.2. Under Assumptions 2.1, 2.5 and 3.1,

Lemma 4 . 1 .

 41 Under Assumption 3.1, with the notations of Corollary 2.6, ifF = F m := {ϕ ∈ S m : ϕ = 1}, then c Fm L(m) 1/2 .Proof. Consider ϕ ∈ S m such that ϕ = 1. Then, there exist a 1 , . . . , a m ∈ R such that

Assumption 4 . 3 .Example 4 . 4 .

 4344 The map m → L(m) has polynomial growth. On the one hand, note that Assumption 4.2 is fulfilled by several usual bases: the trigonometric basis, Hermite's basis, Laguerre's basis, etc. On the other hand, for instance, the trigonometric basis fulfills Assumption 4.3 (see Example 3.3). Now, let us establish the main result of this section: an oracle inequality for f m . Theorem 4.5. Under Assumptions 2.1, 2.5, 3.1, 4.2 and 4.3, if K 16c 2.3 , then there exist two positive constants c 4.5,1 and c 4.5,2 , not depending on n, such that

2 + 2 -

 22 |ν n (ϕ)| -h(m, m) 1/2 2h(m, m) -p(m, m) + = 2 sup ϕ∈Fm∨m |ν n (ϕ)| -h(m, m) 1/2 p(m, m) |ν n (ϕ)| -h(m, m) 1/2 > (r + p(m, m )) 1/2 dr.

Figure 1 .

 1 Figure 1. Projection estimations of f computed for m = 30 on 10 datasets with H = 0.5.

Figure 2 .

 2 Figure 2. Projection estimations of f computed for m = 35 on 10 datasets with H = 0.7.

† Laboratoire Modal'X, Université Paris Nanterre, Nanterre, France Email address: nmarie@parisnanterre.fr


Mohammed Akram Fellah
email: mafellah@uwaterloo.ca

Mohammed Akram

Fellah Asymptotic

Asymptotic Methods


domain: sde phys

text:
Simulation protocoles

Each model under the responsability of one particular team with 3 successive tasks.

1.

Simulate observed yields and water use at 4 sites with a minimum of local data: cv phenology, soil, weather, techniques.

2.

Adjust parameters with all experimental data on yields, LAI, nitrogen etc… Feedback of CC impact on Maize environmental variables for wheat and maize

Figure 2 .

 2 Figure 2. (a) Comparaison des réponses de l'assimilation nette maximale de CO 2 en fonction de la teneur en CO 2 atmosphérique ([CO 2 ]) entre 0 et 700 ppm, en conditions optimales de lumière et d'alimentation minérale et hydrique, pour les plantes en C3 et en C4 ; d'après Saugier (1983) ; (b) réponse à la teneur en CO 2 de la conductance stomatique normée à sa valeur prise aux concentrations standard au moment de l'expérience. Cette réponse est identique pour les plantes en C3 et C4 ; d'après Francks et al. (2013).

  Martreet al. 2015. Multimodel ensembles of wheat growth: many models are better than one. Global Change Biology.

Fig. 1 .

 1 Fig. 1. Modeled average 30-yr changes in (A) yield, (B) transpiration, (C) soil nitrate, and (D) soil organic carbon (SOC) as a function of mean temperature increase over the range 0, +3°C, +6°C, under [CO2] baseline conditions (360 ppm) for wheat (empty bars: AR, Argentina; AU, Australia; IN, India; NL, the Netherlands) and maize sites (filled bars: BR, Brazil; FR, France; TZ, Tanzania; US, United States). For each boxplot, the black dot represents the median (50th percentile), while the bars span 25th to 75th percentiles and lines span 10th to 90th percentiles of model ensemble results.