Emilie Bosc

Vincent Jantet

Muriel Pressigout

Luce Morin

Christine Guillemot

BIT-RATE ALLOCATION FOR MULTI-VIEW VIDEO PLUS DEPTH

Keywords: 3DTV, MVD, 3D video, MVC, quality assessment

The efficient compression of multi-view-video-plus-depth (MVD) data raises the bit-rate allocation issue for the compression of texture and depth data. This question has not been solved yet because not all surveys reckon on a shared framework. This paper studies the impact of bit-rate allocation for texture and depth data relying on the quality of an intermediate synthesized view. The results show that depending on the acquisition configuration, the synthesized views require a different ratio between the depth and texture bit-rate: between 40% and 60% of the total bit-rate should be allocated to depth.

INTRODUCTION

Multi-view video plus depth (MVD) data is a set of multiple sequences capturing the same scene at different viewpoints, with their associated per-pixel depth value. MVD data lead to two main applications: 3D television (3DTV) that provides a depth feeling, and Free Viewpoint Video (FVV), that allows navigation inside the scene. Depth data provide information on scene geometry and help in virtual intermediate view generation. Since texture and depth information are required for view synthesis, an efficient coding framework should ensure the preservation of essential data. Indeed, previous studies ( [START_REF] Merkle | Multi-view video plus depth representation and coding[END_REF], [START_REF] Vetro | Towards a 3D video format for auto-stereoscopic displays[END_REF]) have shown that coding artifacts on depth data can dramatically influence the quality of the synthesized view. Depth maps are not natural images. Most of the state-of-the-art used codecs for depth maps are inspired from 2D video codecs that are optimized for human visual perception of color images. Yet, a straightforward idea suggests that being a monochrome signal, depth maps require low bit-rate compared to texture data. Actually, because of its capital role in the view synthesis processing, compression artifact of such data may lead to fatal synthesis errors when generating virtual views. Consequently, a simple but essential question refers to the bit allocation ratio between texture and depth. This rate ratio depends on the targeted application. Here, we address this question by measuring the Peak Signal to Noise Ratio (PSNR) scores of intermediate views which need to be generated in contexts of 3DTV (for ren-dering on autostereoscopic displays) or of FVV for rendering view points different from those captured by the cameras. The use of this objective metric is justified by its simplicity and mathematical easiness to deal with such purposes, although previous studies [START_REF] Tikanmaki | Quality assessment of 3D video in rate allocation experiments[END_REF] highlighted the need for new metrics for 3D video. The appropriate rate ratio that should be used is not clearly stated in the literature: most of the studies do not rely on the same framework. Fehn et al. [START_REF] Fehn | Depth-image-based rendering (DIBR), compression and transmission for a new approach on 3D-TV[END_REF] show that being a gray-scale signal, the depth video can be compressed more efficiently than the texture video using less than 20% of the texture bit-rate for video-plus-depth data format. In [START_REF] Liu | Joint video/depth rate allocation for 3D video coding based on view synthesis distortion model[END_REF], the authors proposed an efficient joint texture/depth rate allocation method based on a view synthesis model distortion, for the compression of MVD data. According the the bandwidth constraints, the method delivers the best quantization parameters combination for depth/texture sets that maximizes the rendering quality of a synthesized view in terms of MSE. Our experiments tried to quantify the appropriate rate ratio between depth and texture data, and then analyze the relationship with the encoded sequence. Section 2 is devoted to virtual view synthesis. Section 3 states the experimental protocol used to assess the influence of the texture/depth compression ratio while Section 4 discusses the results. Finally Section 5 concludes the paper.

VIRTUAL VIEW SYNTHESIS

For 3DTV or FVV, the transmitted texture and depth sequences are used to generate virtual views with the help of depth-image-based rendering techniques. The generated views can then be rendered on a conventional display, or a stereoscopic or an autosterescopic display. Generating a "virtual" view consists in synthesizing a novel view of the scene, from a viewpoint which differs from those captured by the cameras, relying on the available texture and depth data. The texture, that is the conventional 2D color sequences, gives the color information. The depth data are gray-scales images and are considered as a monochromatic signal. Each pixel of a depth image, also called depth map, indicates the distance of the corresponding 3D-point from the camera. Based on projective geometry [START_REF] Hartley | Multiple view geometry in computer vision[END_REF], the 3D representation of a scene can be retrieved from a depth map. The presented experiments are based on the View Synthesis Reference Software (VSRS) [START_REF] Tanimoto | Reference softwares for depth estimation and view synthesis[END_REF], version 3.5, provided by MPEG. Figure 1 illustrates the relationship between a real 3D point of the scene, defined as X and its projections x 1 and x 2 in camera planes of C 1 and C 2 respectively. The geometric transformation from 3D world to the camera plane can be easily performed from depth data, and both intrinsic and extrinsic parameters of known cameras. On the same principle, 3D points of the real world can be projected onto the image plane of a virtual camera from an arbitrary viewpoint. VSRS is able to deal with such cases, providing it the parameters related to the virtual camera C v and texture and depth information from two adjacent known views. Figure 2 shows the principles of synthesis used in VSRS. Depth maps of the two adjacent views are projected into the target view resulting in two new depth maps referring to the virtual viewpoint. Those new maps contain non-valued areas, called holes. They correspond to occluded areas in the reference viewpoint (left or right respectively). Consequently, when the left and right texture images are projected in the target viewpoint according to the new depth maps, they also contain non-valued areas. Those areas are then filled in by available information from both new texture images. Then the two texture images are fused into one single image that is usually post-processed by in-painting methods. The described process assigns each pixel of the new texture image T v , a color value according to its corresponding depth. Three cases are considered:

-both depth values for the considered pixel are null: this is an non visible area.

-only one of the two pixels has a depth value: this is a occluded area in one of the reference viewpoints.

-depth values of the pixels in the adjacent views are not null. This is expressed by:

Tv =                      0, if (u, v) is not visible T 1 (u, v), if d 1 (u, v) = 0 and d 2 (u, v) = 0 T 2 (u, v), if d 1 (u, v) = 0 and d 2 (u, v) = 0 (1 -α)T 1 (u, v) + αT 2 (u, v), if d 1 (u, v) = 0 and d 2 (u, v) = 0
where (u, v) refers to the coordinates of a pixel of the synthesized view, d 1 (u, v) is the depth value of this pixel from camera C 1 , d 2 (u, v) is the depth value of this pixel from camera C 2 , and α is a factor depending on the distance (α < 1).

The synthesis process already raises some issues. First, in terms of geometry: occluded regions lead to non-valued areas in the texture image. Secondly, errors can occur because pixel coordinates do not locate at an integer position and is usually either interpolated or rounded to the nearest integer position. In-painting methods as well as interpolation filters are developed in order to reduce the synthesis artifacts.

PROTOCOL

Our aim was to evaluate the required ratio between depth and texture data relying on the quality of a reconstructed view, in terms of PSNR. Thus, we used the Multiview Video Coding (MVC) reference software, JMVM 8.0 (Joint Multiview Video Model) to encode three views, as a realistic simulation of a 3DTV use. To vary the bit-rate ratio and the total bit-rate, the quantization parameter QP varies from 20 to 44 for both depth and texture coding. The central view predicts the two other views. Then, from the decompressed views, we computed the intermediate view between the central view and the right one, by using the reference software: VSRS, version 3.5, provided by MPEG. Figure 3 illustrates the described protocol. We used two different types of sequences to answer our question: Ballet from Microsoft Research, and Book Arrival from Fraunhoffer HHI (1024 × 768). This last sequence is a 3DV test material in MPEG. It was acquired through cameras arranged equidistantly along a straight line with a rectified configuration (no gap in-between the cameras and the interaxial distance is 65 mm). On the other hand, Ballet was acquired with converging cameras. The encoded views are 2, 4 and 6 for Ballet, and 6, 8 and 10 for Book Arrival. For each couple (QP texture , QP depth ), the average PSNR score of the synthesized sequence is evaluated, compared to the original acquired view.

RESULTS AND DISCUSSION

Figure 4 presents the results. The average PSNR of the synthesized sequence are plotted over the bit-rate percentage assigned to depth. The different curves correspond to interpolation of the measured points for different ranges of bit-rate. We observe that for a given sequence, no matter the bit-rate, the ratio that provides the best quality is the same: it seems to be around 60% for Ballet, and around 40% for Book Arrival. This suggests that the required depth information that enables a good reconstruction quality, in terms of PSNR depends on the content. More precisely, we think that this percentage depends on the acquisition configuration of the sequence, i.e. the camera baseline. The synthesized sequence from Book Arrival may require less elements of depth for the reconstruction because of the linear configuration of the three used cameras. On the contrary, the synthesized sequence from Ballet, seems to require an important amount of depth information to ensure a good quality of reconstruction. Correct reconstruction around dis-occluded areas may require more reliable depth information depending on the reference camera position. These results are consistent with [START_REF] Liu | Joint video/depth rate allocation for 3D video coding based on view synthesis distortion model[END_REF]: although the authors do not state clearly the appropriate ratio for those two sequences, their rate/distortion curves show that, for example, the bit-rate pair (962kbps for texture, 647kbps for depth), i.e. a percentage of 40% for depth, gives better synthesis quality (in terms of PSNR) for Book Arrival. The synthesis condi- tions are similar to our experiments. On the other hand, in [START_REF] Fehn | Depth-image-based rendering (DIBR), compression and transmission for a new approach on 3D-TV[END_REF], the synthesis conditions involves video-plus-depth data: in this case, a very little continuum is supported around the available original view. Since synthesis distortion increases with the distance of the virtual viewpoint, this explains the significant difference with our results.

Figure 5 shows particular areas of the synthesized views from the presented experiment. Figures 5(a), 5(d) and 5(g) show that allocating less than 10% of the total bit-rate to depth data induces important damages along the edges. The location of the depth map discontinuities is deeply compromised which leads to errors in the synthesized view. PSNR scores fall down because of the numerous errors along the contours of objects. Figures 5(c), 5(f) and 5(i) suggest that affecting more than 80% of the total bit-rate to depth data preserves the edges of some objects but texture information is lost because of the coarse quantization. Affecting between 40% and 60% of the total bit-rate to depth data seems to be a good tradeoff for the tested sequences, as it can be observed in Figure 5(b), 5(e) and 5(h). PSNR scores and visual quality are both improved compared with the two other presented cases. The depth maps are accurate enough to ensure correct projections and decompressed texture images are good enough to avoid drastic artifacts.

CONCLUSION

This paper aimed at determining the appropriate ratio for joint depth/texture compression, in the MVD framework. The experiment consisted in encoding both texture and depth data by the same compression scheme, that is to say using the MVC coder. The attributed depth ratio was varied from 2% to nearly 95% and the synthesis of an intermediate view was performed. The average PSNR score of the synthesized views was then evaluated. As expected, the compression of texture data induces blurred regions in the synthesized views, while the coarse compression of depth data leads to geometrical errors during the pixels projection. An ideal ratio between depth and texture can be found in order to maximize the quality of the synthesized view; this ratio seems to be the same for a given sequence no matter the total rate. The obtained results showed that the best quality of reconstruction by using VSRS may require to affect between 40% and 60% of the total bit-rate to depth data, depending on the available MVD data. Those observations are related to H.264/MVC encoding. Using a different encoding framework may allow to reduce the ratio for depth. In future work, the analysis of the MVD data and related parameters such as video contents and camera settings should be studied to automatically set the correct ratio between depth and texture according to these parameters.

Fig. 1 .

 1 Fig. 1. Relationship between image points and real world [8].

Fig. 2 .

 2 Fig. 2. View synthesis with VSRS.

Fig. 3 .

 3 Fig. 3. Experimental protocol.

  (a) PSNR (dB) of synthesized views as a function of rate allocated to depth in percentage of total rate for Ballet (b) PSNR (dB) of synthesized views as a function of rate allocated to depth in percentage of total rate for Book Arrival

Fig. 4 .

 4 Fig. 4. Interpolated rate-distortion curves of synthesized views.

  (a) PSNR = 30.0dB; Depth = 3% of bit-rate; (b) PSNR = 33.8dB; Depth = 60% of bit-rate; (c) PSNR = 30.8dB; Depth = 95% of bit-rate; (d) PSNR = 36.96dB; Depth = 6% of bit-rate; (e) PSNR = 39.38dB; Depth = 38% of bit-rate; (f) PSNR = 34.17dB; Depth = 88% of bit-rate; (g) PSNR = 36.96dB; Depth = 6% of bit-rate; (h) PSNR = 39.38dB; Depth = 38% of bit-rate; (i) PSNR = 34.17dB; Depth = 88% of bit-rate;

Fig. 5 .

 5 Fig. 5. Synthesized images from MVD data, with different bit-rate ratios between texture and depth.

ACKNOWLEDGMENTS

This work is supported by the French National Research Agency as part of PERSEE project (ANR-09-BLAN-0170) and CAIMAN project (ANR-08-VERS-002). We would like to acknowledge the Interactive Visual Media Group of Microsoft Research for providing the "Ballet" data set, Fraunoffer HHI for providing the "Book Arrival" data set and MPEG for providing VSRS.


Sandrine Faure

Pascal De Santa

Pascal De

Santa Barbara
email: barbara@inserm.fr

Molecular embryology of the foregut

Keywords: Animals, Body Patterning, Cell Line, Disease Models, Animal, Epithelial-Mesenchymal Transition, Epithelium, embryology, Esophageal Atresia, embryology, Esophagus, embryology, Gene Expression Regulation, Developmental, Genes, Homeobox, Humans, Lung, embryology, Mice, Mice, Knockout, Mutation, Tracheoesophageal Fistula, embryology, Upper Gastrointestinal Tract, embryology

teaching and research institutions in France or abroad, or from public or private research centers.

Author Keywords foregut ; Shh ; Bmp ; adriomycin ; endodermal-mesenchymal interaction.

The digestive and respiratory systems have different physiological functions (nutrition, food transit and evacuation for the first; breathing and oxygen supply to the blood for the second) and are generally considered and studied as two independent structures.

Nevertheless, although at birth they are separated, they both derive from a common and transient developmental structure, the foregut, which is the anterior part of the gastrointestinal (GI) tract ( ). The GI tract is a remarkably complex, three-dimensional, specialized and 1

vital system that is derived from a simple tube-like structure. The vertebrate GI tract includes the luminal digestive system (i.e., esophagus, stomach, intestine and colon, which we will designate on the whole as gut ) and the GI-tract derivatives (i.e., thyroid, lungs, liver and " " pancreas). The gut is composed of three germ layers: mesoderm (which forms the smooth muscle layer), endoderm (which forms the epithelial lining) and ectoderm (which includes the enteric nervous system). The gut develops from two invaginations at the anterior (anterior intestinal portal, AIP) and posterior (caudal intestinal portal, CIP) end of the embryo that elongate and fuse to form a straight tube. The primitive gut tube is initially patterned into three broad domains along its anterior-posterior (AP) axis: the fore-, mid-and hindgut. As they develop, each region of the gut is characterized by unique mesodermal and endodermal morphologies, which can easily be discerned by gross and microscopic examination. Specifically, these tissues show regional differentiation along the AP axis that specifies pharynx, esophagus and stomach (the foregut), small intestine (the midgut) and large intestine (hindgut). This regionalization is maintained throughout life and is essential and necessary for normal gut function. These patterning events are remarkably conserved across species ( ) and patterning anomalies are likely to be responsible for many of the human gut malformation syndromes, such as 1 tracheo-esophageal atresia (TE), infantile hypertrophic pyloric stenosis or anal atresia ( , ). 1 2

Foregut development

Lung and gut are two independent systems that originate from one common embryonic organ, the foregut. The development of the foregut is not well documented in comparison to that of other parts of the digestive system. The respiratory system originates from the formation of an endodermal diverticulum in the ventral wall of the foregut, whereas the esophagus forms from the foregut dorsal wall ( ). 1

Specifically, the foregut endoderm evaginates and pushes the surrounding mesenchyme to form the two presumptive lung buds. In avian embryos, these processes manage to form two independent and separate endodermal structures, dorsally the esophagus and ventrally the lung buds, in a very short time (around 10 hours). Later on, the lung buds grow caudally, leading to the formation of a temporary TE septum. The appearance of the TE septum is followed first by the expansion of the trachea and by the final separation of the two endodermally-derived systems.

Molecular pathways involved in foregut development

The cellular events that contribute to foregut specification, lung development and separation of trachea and esophagus are regulated by different and specific molecular pathways. The regionalization of the different parts of the gut is controlled by the localized expression of different genes, which are homeobox-containing transcription factors ( ). For example, and are specifically expressed Hox 3 Hoxa3 Hoxb4

in the foregut endoderm, whereas and are expressed more caudally, respectively in the midgut and hindgut endoderm ( ,

Hoxc5 Hoxa13 3 4
). Sonic Hedgehog (Shh), a member of the Hedgehog (Hh) family of morphogens is expressed in the entire endodermal layer but not in the pancreas ( ). Moreover, Ligtingting and colleagues reported a specific and dynamic expression pattern of during TE development (also called TTF1 or T/EBP) is specifically expressed in the anterior part of the foregut and in the endoderm of the developing trachea but not of the esophagus ( ). All these genes are expressed in the foregut endoderm in a spatially regulated manner ( ) and therefore 8

Figure 1 their deregulation could be implicated in the genesis of TE malformations.

In order to investigate their function(s) and requirement during foregut development, these genes have been inactivated in different transgenic mouse lines. For instance, inactivation of is associated with the presence of a common TE lumen suggesting that Nkx2.1

NKX2

Epithelial-mesenchymal interactions during the foregut development

Epithelial-mesenchymal interactions are essential for the development and differentiation of the GI tract ( ). However, the 7 contribution of the mesenchyme during the development of the foregut is not often commented. Different studies reported a significant mesenchymal condensation at the site of TE separation ( , ). Moreover, in rats, treatment with the teratogenic agent adriamycin 10 11

reproducibly induces esophageal atresia/TE fistula. Such malformations are associated with lower cellularity and disorder of the surrounding mesenchymal cells already at early stages of foregut misdevelopment ( ), supporting the notion of a major involvement of 11 the foregut mesenchyme during foregut development.

As previously commented, the Hh signaling pathway regulates foregut development through epithelial-mesenchymal interactions and different members of this pathway, such as the transcription factor ( ), are expressed in the foregut mesenchyme. To analyze the Gli2 12 interplay of the different components of the Hh pathway that are expressed in the foregut mesenchyme, Motoyama and colleagues inactivated in mice and observed the presence of TE malformation associated with alterations of visceral smooth muscle cells ( ). In Gli2 13 addition, inactivation of both and caused a stronger phenotype with a common TE structure, demonstrating that inhibition of Gli2 Gli3

genes of the Hh pathway, which are normally expressed in the foregut mesenchyme, triggers similar phenotypes as inactivation of the endodermally expressed . shh Recently, also the function of the BMP signaling pathway during TE development was investigated by the Hogan s laboratory. This

Conclusion and perspectives

Candidate factors for foregut development include known pattern formation genes that were first identified in , such as Drosophila

3

 3 shh with earlier expression and patterning of the ventral foregut by Shh and transient inhibition of expression in the tracheal endoderm ( ). shh 5 Others transcription factors also show specific expression profiles in the endoderm, such as the SRY-related High-Mobility Group (HMG) transcription factors (foregut endoderm) and (midgut/hindgut endoderm) ( , ). The homeodomain transcription factor Sox2

'

  signaling pathway is activated in the gut mesenchyme following expression and activation of and . While is localized to the shh Patch shh ventral foregut endoderm, is expressed in the neighboring mesenchyme but not in the dorsal part of the foregut, which will give rise Bmp4 to the esophagus ( ) and where the inhibitor of the BMP pathway, is expressed. To analyze the function of the BMP pathway 14 Noggin, during foregut development, Que and colleagues inactivated and observed the presence of a common TE structure ( ). In a Noggin 14 rescue experiment, where / mice where crossed with / mice to decrease the activity of the BMP pathway, the progeny Noggin --BMP4 + -showed normal separation and formation of trachea and esophagus, supporting the idea that the level of BMP activity is crucial for the development of these two structures. These examples demonstrate that the foregut mesenchyme is essential for the correct development of the foregut and that the mesenchyme could actively participate in the development of these two structures.

  nuclear homeotic transcription factors (HOX, SOX and NKX factors) and secreted factors (BMP and Hh factors) ( , , , ). Genetic 3 6 15 16 evidences from different animal models indicate that these molecules play multiple and crucial roles in the development and septation of trachea and esophagus. The identification of and (CDH) mutations in syndromic NMYC chromodomain helicase DNA binding CDH7 forms of TE bring us new factors that could be involved during the development of the trachea and esophagus ( ). To better understand 16 the molecular basis of human malformations and syndromes, such as TE atresia, it is essential to better dissect normal foregut development at the molecular level and to stimulate translational research between scientists and clinicians.

Figure 1

 1 Figure 1 Molecular pathways involved in the regionalization of the foregut endoderm. Expression boundaries of selected factors expressed in the endoderm along the antero-posterior axis. The embryonic GI tract is divided into fore-, mid-and hindgut. ( ) is expressed Sonic hedgehog shh in the entire endodermal layer sparing the pancreas. genes which are homeobox-containing transcription factors harbor a specific Hox localized expression into the GI endoderm. SRY-related High Mobility Group (HMG) transcription factors and present exclusive Sox2 Sox9 pattern in GI endoderm. The transcription factor is expressed into the foregut. Adapted from Zorn and Wells (2009) ( ). Nkx2.1 15

  common TE lumen, demonstrating an essential function of Shh during TE development. Shh is a ligand and activator of the Hh signaling pathway. During the development of the GI tract, is localized to the gut endoderm, but it acts on the gut mesenchyme to control shh endodermal-mesenchymal interactions ( ). After processing by the Golgi apparatus, Shh is secreted by endodermal cells and then binds to 3 its receptor PATCHED (or PTCH1), which is expressed in the surrounding mesenchymal cells. PATCHED activation is followed by cleavage and activation of the GLI1/2 transcription factors, which regulate various genetic networks and the expression of

	mesenchymal-specific genes, such as	, Ptch1 Gli1	and	( Bone Morphogenetic Protein 4 Bmp4 3 7 ) ( , ).

.1 is essential for the development and separation of TE endoderm ( ). Inactivation of in mice, like for leads to a 9 shh Nkx2.1,

Acknowledgements:

Supported by grants from Agence Nationale pour la Recherche (ANR-07-JCJC-0112), Association Fran aise contre les Myopathies, R gion


Thomas M Michelitsch

Gérard A Maugin

Mujibur Rahman

Shahram Derogar

Andrzej F Nowakowski

Franck C G A Nicolleau

A self-similar field theory for 1D linear elastic continua and self-similar diffusion problem

Keywords: fractal mechanics, fractional calculus, generalized functions, distributions, self-similarity, selfsimilar Laplacian, Green's functions, statics, dynamics, diffusion

This paper is devoted to the analysis of some fundamental problems of linear elasticity in 1D continua with self-similar interparticle interactions. We introduce a self-similar continuous field approach where the self-similarity is reflected by equations of motion which are spatially non-local convolutions with power-function kernels (fractional integrals). We obtain closed-form expressions for the static displacement Green's function due to a unit δ-force. In the dynamic framework we derive the solution of the Cauchy problem and the retarded Green's function. We deduce the distribution of a self-similar variant of diffusion problem with Lévi-stable distributions as solutions with infinite mean fluctuations describing the statistics Lévi-flights. We deduce a hierarchy of solutions for the self-similar Poisson's equation which we call "selfsimilar potentials". These non-local singular potentials are in a sense self-similar analogues to the 1D-Dirac's δ-function. The approach can be the starting point to tackle a variety of scale invariant interdisciplinary problems.

INTRODUCTION

The discovery of fractal geometry in nature by Benoît Mandelbrot in the seventies of the last century had a great impact on science and opened us a new view on nature [START_REF] Mandelbrot | The Fractal Geometry of Nature[END_REF][START_REF] Mandelbrot | Fractales, hasard et finance[END_REF][START_REF] Mandelbrot | Les objets fractals -form hasard et dimension[END_REF][START_REF]Universalités et Fractales[END_REF]. In the last decade also an increasing interest has risen in the mechanics of materials having a microstructure which is exactly or statistically invariant over a wide range of scales. Such materials can be in a good approximation idealized as "fractal" materials. Such an idealization is to be understood in the same way as the notion of the "infinite medium". Despite there is no infinite medium in nature, this notion has proven to be a highly appropriate idealization. Some interesting results to describe the mechanical properties in fractal materials have been achieved recently [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF][START_REF] Bondarenko | Self Similar Spectrum of Fractal Lattice[END_REF][START_REF] Ostoja-Starzewski | Towards a thermomechanics of fractal media[END_REF][START_REF] Epstein | The stiffness of self-similar fractals[END_REF]. However a generally accepted theory of "fractal mechanics" is not yet established so far. Therefore simple even simplistic models capturing only a few aspects of the characteristic fractal behavior in materials are highly desirable.

The point of departure in the present paper is the continuum limit of the 1D medium which has been defined as "quasi-continuous linear chain" in [START_REF] Michelitsch | Dispersion relations and wave operators in self-similar quasicontinuous linear chains[END_REF]. Before we start with our demonstration we should like to specify "our" notion of "self-similarity": The notion self-similarity used in the present paper as well as in [START_REF] Michelitsch | Dispersion relations and wave operators in self-similar quasicontinuous linear chains[END_REF] corresponds to the notion of "self-similarity at a point" commonly used in the mathematical literature [START_REF] Peitgen | Fractals for the Classroom: Part 1: Introduction to Fractals and Chaos[END_REF]. An object is self-similar in the strict sense if it consists of parts which are exact rescaled copies of the entire object. In contrast is the notion "self-similarity at a point" (with "point 1 we mean a space-point which is contained by or attached to the object, where we place for example the origin of a coordinate system): An object which is "self-similar at a point" contains only a single part which is a re-scaled copy of the entire object [START_REF] Peitgen | Fractals for the Classroom: Part 1: Introduction to Fractals and Chaos[END_REF]. The notion of "self-similarity at a point" implies that the scaling invariance repeats over an infinity of scales. In this paper as well as in [START_REF] Michelitsch | Dispersion relations and wave operators in self-similar quasicontinuous linear chains[END_REF] we call an object just "self-similar" without always mentioning that we actually mean "self-similar at a point". To be more precise: When we call a function Λ(h) self-similar with respect to h which we define by Λ(N h) = N δ Λ(h) for a prescribed scaling factor N , then strictly speaking the function Λ(h) is self-similar at the point h = 0. The simplest self-similar functions of this type are power-functions h δ .

The paper is organized as follows: In section 3 we deduce static Green's function of displacements due to a δ-unit force in closed form. In section 4 we construct the basic solution of the Cauchy problem in the form of two integral kernels which give the solutions due to prescribed initial displacement-and velocity-field, respectively, in analogy to the classical d'Alembert's solution. One of these two kernels determine also the retarded time-domain dynamic Green's function.

In section 5 we analyze a self-similar diffusion problem which we formulate as an initial value problem for a given initial distribution by utilizing our self-similar Laplacian. The solutions of this problem turn out to be Lévi-stable distributions with infinite mean fluctuations. As a spin-off result we obtain a hierarchy of functions which we call "self-similar potentials" as they play the analogue role for our self-similar Laplacian as the Newtonian potentials do in the case of a "traditional" Poisson-equation. These hierarchy of potentials includes also the static Green's function as a solution of a δ-type source.

The self-similar elastic continuum

We consider a purely elastic medium in 1D with a spatial continuous constant mass distribution of density 1 where any spatial point x represents a material point which interacts harmonically with an ensemble of other material points located at x ± hN s (s ∈ Z 0 ) in such a way that the elastic energy-density at x is a self-similar function with respect to h (at point h = 0). This is expressed below by relation [START_REF] Mandelbrot | Les objets fractals -form hasard et dimension[END_REF]. The fractal properties of the dispersion relation of such a system have been considered in our recent paper [START_REF] Michelitsch | Dispersion relations and wave operators in self-similar quasicontinuous linear chains[END_REF]. As a point of departure we invoke some results from that paper: The Hamiltonian of that material system is

H = 1 2 ∞ -∞ u2 (x, t) + V(x, t, h) dx (1)
where x denotes the space-and t the time coordinates. V(x, t, h) indicates the elastic energy density2

V(x, t, h) = 1 2 ∞ s=-∞ N -δs (u(x, t) -u(x + hN s , t)) 2 + (u(x, t) -u(x -hN s , t)) 2 (2)
which converges in the range 0 < δ < 2 and where we assume h > 0 and N being a prescribed scaling factor. Without loss of generality we can restrict ourselves to N > 1 (N ∈ R). In (1), (2) u and u = ∂ ∂t u stand for the displacement field and the velocity field, respectively. (2) has the property of being self-similar with respect to h at point h = 0, namely

V(x, t, N h) = N δ V(x, t, h) (3)
As a point of departure for the approach to be developed we evoke the continuum limit of (2) and the resulting equation of motion. We define the continuum limit as N = 1 + ζ (0 < ζ << 1) so τ = hN s becomes a continuous variable and we can write a self-similar function Λ(h) which fulfills a self-similarity condition (3) asymptotically as [START_REF] Michelitsch | Dispersion relations and wave operators in self-similar quasicontinuous linear chains[END_REF]

Λ(h) = ∞ s=-∞ N -δs f (N s h) ≈ h δ ζ ∞ 0 f (τ ) τ δ+1 dτ (4)
having the form of a power function Λ(h) = const h δ . Both the discrete as well as the continuous representation of (4) converge for sufficiently good functions (see details in [START_REF] Michelitsch | Dispersion relations and wave operators in self-similar quasicontinuous linear chains[END_REF]). From (4) follows that in that continuum limit we can write the elastic energy density (2) as a functional of the displacement field u(x, t) in the form

V(x, t, h) = h δ 2ζ ∞ 0 (u(x, t) -u(x + τ, t)) 2 + (u(x, t) -u(x -τ, t)) 2 τ δ+1 dτ ( 5
)
which exists as in the discrete case (2) in the band 0 < δ < 2. The equation of motion (self-similar wave equation) has then the form [13]

∂ 2 ∂t 2 u(x, t) = ∆ (δ,h,ζ) u(x, t) (6)
The right hand side of (6) can be conceived as the self-similar Laplacian of the medium. The self-similar Laplacian is necessarily a non-local and self-adjoint negative definite operator and is obtained as

∆ (δ,h,ζ) u(x) = h δ ζ ∞ 0 (u(x -τ ) + u(x + τ ) -2u(x)) τ 1+δ dτ , 0 < δ < 2 (7)
and exists in the interval 0 < δ < 2. It might be sometimes convenient to rewrite [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF] in the equivalent form

∆ (δ,h,ζ) u(x) = h δ ζδ d dx ∞ 0 (u(x + τ ) -u(x -τ )) τ δ dτ , 0 < δ < 2 (8)
We can express Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF], [START_REF] Bondarenko | Self Similar Spectrum of Fractal Lattice[END_REF] in terms of Weyl-Marchaud fractional derivatives which are defined as [START_REF] Zähle | Fractional derivatives of Weierstrass-type functions[END_REF]

D δ l u(x) = δ Γ(1 -δ) ∞ 0 (u(x) -u(x -τ ) τ δ+1 dτ , left -sided , 0 < δ < 1 (9)
and

D δ r u(x) = (-1) δ δ Γ(1 -δ) ∞ 0 (u(x) -u(x + τ ) τ δ+1 dτ , right -sided , 0 < δ < 1 ( 10
)
where 0 < δ < 1 indicates the range of existence of each fractional integral D δ l,r u(x), respectively. Γ(z) indicates the Γ-function (faculty-function) [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] Γ

(z + 1) =: z! = ∞ 0 e -τ τ z dτ, Re(z) > -1 (11)
The condition Re(z) > -1 is required that integral [START_REF] Ross | Functions that have no First Order Derivative might have Fractional Derivatives of all Orders Less than One[END_REF] exists. Then the Laplacian (7) has the representation3

∆ (δ,h,ζ) u(x) = - h δ Γ(1 -δ) ζδ D δ l + (-1) -δ D δ r u(x) , 0 < δ < 1 (12)
where the actual range of existence of ( 12) is 0 < δ < 2 when we write both terms as one single integral like in [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF]. We should like to mention that the representation [START_REF] Zähle | Fractional derivatives of Weierstrass-type functions[END_REF] in terms of standard fractional integrals is not crucially helpful in order to solve problems. Therefore, we should rather conceive Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF], [START_REF] Bondarenko | Self Similar Spectrum of Fractal Lattice[END_REF] itself as definition of a self-adjoint combination of fractional derivatives. The goal of this paper is to analyze some static and dynamic problems which are governed by the selfsimilar Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF], [START_REF] Bondarenko | Self Similar Spectrum of Fractal Lattice[END_REF]. The following section is devoted to deduce an explicit expression for the static displacement Green's function.

Static Green's function

This paragraph is devoted to deduce the static Green's function for the displacement field due to a δ-point force. We define this Green's function g(x) by

∆ (δ,h,ζ) g(x) + δ(x) = 0 ( 13
)
where δ(x) denotes Dirac's δ-function. The static displacement field u(x) due to a force density f (x) is then defined by

∆ (δ,h,ζ) u(x) + f (x) = 0 (14)
and can be represented as a convolution

u(x) = ∞ -∞ g(x -τ )f (τ )dτ (15)
with the static Green's function g(x) of ( 13) as convolution-kernel. By introducing the Fourier transforms

g(x) = 1 2π ∞ -∞ g(k)e ikx dk ( 16
)
and

δ(x) = 1 2π ∞ -∞ e ikx dk (17)
it follows from (13) that

g(k) = 1 ω 2 (k) (18)
By taking into account that e ikx are the eigenfunctions of the ∆ (δ,h,ζ) -operator [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF] with ∆ (δ,h,ζ) e ikx = -ω 2 (k)e ikx the dispersion relation is obtained from

ω 2 (k) = - h δ ζ ∞ 0 e ikτ + e -ikτ -2 τ -δ-1 dτ ( 19
)
which was also given in [START_REF] Michelitsch | Dispersion relations and wave operators in self-similar quasicontinuous linear chains[END_REF] in the form

A δ = 2h δ ζ ∞ 0 (1 -cos(s)) s 1+δ ds , 0 < δ < 2 (20)
This integral exists in the same interval 0 < δ < 2 just as the Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF]. We use now the identity

4 |τ | -δ-1 = 1 cos π(δ+1) 2 Re{ lim ǫ→0+ (ǫ -iτ ) -δ-1 } = lim ǫ→0+ 1 δ! cos π(δ+1) 2 Re{ ∞ 0 e -s(ǫ-iτ ) s δ ds} (21)
which is well defined in the range 0 < δ < 2 of existence of the Laplacian. Replacing identity [START_REF] Michelitsch | Dynamic Eshelby tensor and potentials for ellipsoidal inclusions[END_REF] in the integral (19) yields5

ω 2 (k) = - h δ π ζδ! cos π(δ+1) 2 ∞ 0 s δ {δ(s -|k|) -2δ(s)} ds = h δ π ζδ! sin πδ 2 |k| δ , 0 < δ < 2 (22)
which has the form [START_REF] Michelitsch | Dispersion relations and wave operators in self-similar quasicontinuous linear chains[END_REF]]

ω 2 (k) = A δ |k| δ , 0 < δ < 2 (23)
where coefficient A δ is given by

A δ = h δ ζ π δ! sin ( πδ 2 ) > 0 , 0 < δ < 2 (24)
We conclude that the positiveness of the constant A δ > 0 (which is determined by sin πδ 2 > 0) of (24) due to the positiveness of integral [START_REF] Jackson | Classical Electrodynamics, Third Edition[END_REF] reflects nothing but the condition of elastic stability, i.e. ω 2 (k) > 0 for k = 0 which is a physically necessary condition. This condition together with the condition of existence of Laplacian (7) constitute a physically sufficient condition which gives the δ-range of physical consistency namely the range 0 < δ < 2. We emphasize that the regularization performed in [START_REF] Michelitsch | Dynamic Eshelby tensor and potentials for ellipsoidal inclusions[END_REF] would allow to obtain converging integrals ∀δ > -1. However among these mathematically admissible δ-values only those within 0 < δ < 2 lead to physically meaningful results. This is true for all physical problems formulated with Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF].

Explicit Green's function:

The integral (16) assumes with ( 18) and ( 23) the form

g(x) = 1 2πA δ ∞ -∞ |k| -δ e ikx dk ( 25
)
where we make use of the relation

|k| -δ = 1 (δ -1)! cos δπ 2 lim ǫ→0+ Re{ ∞ 0 e -τ (ǫ-k) τ δ-1 dτ } ( 26
)
which is well defined in the range of existence 0 < δ < 2 of the Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF]. Using this identity to evaluate (25) we finally arrive at

g(x) = g 0 |x| δ-1 (27)
with the pre-factor

g 0 = 1 2A δ (δ -1)! cos πδ 2 = ζδ 2πh δ sin δπ 2 cos δπ 2 , 0 < δ < 2 (28)
Expression (27) holds for 0 < δ < 2 which yields physical consistent behavior for the deformation d dx g(x) having the exponent δ -2 < 0:

| d dx g(x → 0)| → ∞ being singular at x = 0 and | d dx g(x → ±∞)| → 0 vanishing for |x| → ∞.
The consistence of expressions ( 27), (28) with ( 18) can be directly verified by performing the back transformation

g(k) = ∞ -∞ g(x)e -ikx dx = 2g 0 lim ǫ→0+ Re{ ∞ 0 |x| δ-1 e -x(ǫ-ik) dx} ( 29
)
which can be rewritten as

g(k) = 2g 0 Γ(δ) cos πδ 2 |k| -δ (30)
This expression must coincide with the expression g(k) = 1 ω 2 (k) of ( 18) with the dispersion relation [START_REF] Gel'fand | Properties and Operations[END_REF]. This leads to the condition that

1 A δ = 2g 0 Γ(δ) cos πδ 2 (31)
which is fulfilled by (28). It is possible to evaluate (25) in the range 0 < δ < 1 directly to arrive at

g(x) = 1 πA δ lim ǫ→0+ Re{ ∞ 0 e -k(ǫ-ix) |k| -δ dk} = (-δ)! πA δ Re(ǫ -ix) δ-1 = Γ(1 -δ) sin πδ 2 πA δ |x| δ-1 (32)
Comparision of g 0 of (32) and (28) lead to the condition that

Γ(δ)Γ(1 -δ) = π sin πδ (33)
which is known as Euler's reflection formula [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]. Consequently the correctness of expressions ( 27), (28) for the Green's function and ( 18) is proven. For the analysis to follow it is useful to generalize the definition of the Γ-function extending it to α < -1 in such a way that for the extended definition of α! Euler-relation (33

) is fulfilled 6 α! = Γ(α + 1) = - π Γ(-α) sin πα (34)
where Γ(-α) is well-defined by [START_REF] Ross | Functions that have no First Order Derivative might have Fractional Derivatives of all Orders Less than One[END_REF] for α < -1. Definition (34) is motivated in appendix 8.4 for α < -1. Equation (34) complements the usual definition [START_REF] Ross | Functions that have no First Order Derivative might have Fractional Derivatives of all Orders Less than One[END_REF] of the Γ-function by extending it to the "forbidden" α-range of (11) α < -1.

We again empasize our previous observation that the Green's function (27)ff. is physically meaningful only in the range 0 < δ < 2 to obtain consistent behavior for the deformation d dx g(x) behaving as |x| δ-2 , such as its vanishing for |x| → ∞ and its singular behavior for x → 0.

Cauchy problem and dynamic Green's functions

In this section we construct some fundamental solutions of dynamic problems: First in paragraph 4.1 we construct the solution of the Cauchy problem and in paragraph 4.2 the causal (retarded) time-domain Green's function. Further we make some brief remarks on the frequency domain representation the dynamic Green's function representing the solution of the self-similar Helmholtz-equation.

Cauchy problem

The Cauchy problem is defined as follows: Construct the displacement field u(x, t) solving (6) which we rewrite for our convenience now in the form

∂ 2 ∂t 2 u(x, t) = -Lu(x, t) (35)
with ∆ (δ,h,ζ) = -L denotes the self-similar Laplacian of ( 7). The operator L is necessarily self-adjoint 7 and positive definite with Le ikx = ω 2 (k)e ikx and the dispersion relation ω 2 (k) > 0 for k = 0 (Eq. ( 23) of the last section 3). Since L is a linear operator we can take advantage of this and employ in the analysis to follow the calculus of linear operators. The displacement field u(x, t) is required to fulfill the following initial condition at t = 0:

u(x, t = 0) = u 0 (x) (36)
with the prescribed initial displacement field u 0 (x) at t = 0. The velocity field ∂ ∂t u(x, t) is required to fulfil the following initial condition at t = 0:

∂ ∂t u(x, t = 0) = v 0 (x) (37)
where v 0 (x) denotes the prescribed initial velocity field at t = 0. (35) with the initial conditions (36) and (37) represents the Cauchy problem which is defined in the domain -∞ < x < ∞ and -∞ < t < ∞. In order to construct the unique solution of the Cauchy problem, it is convenient to write the displacement field u(x, t) at time t in the form u(x, t) = cos (L

1 2 t) u 0 (x) + L -1 2 sin (L 1 2 t) v 0 (x) (38)
and as a consequence of (38

) the velocity field v(x, t) = ∂ ∂t u(x, t) at time t v(x, t) = -L 1 2 sin (L 1 2 t) u 0 (x) + cos (L 1 2 t) v 0 (x) (39)
Expressions (38), and (39) are defined by their power-series involving only entire powers

L n with n = 0, 1, 2, .. ∈ N 0 of operator L = -∆ δ,h,ζ with the identical operator L 0 = 1.
To evaluate expression (38) it is convenient to write it in its spectral representation which is in our case the Fourier-representation. Introducing the Fourier transformations

u 0 (x) = 1 2π ∞ -∞ ũ0 (k)e ikx dk ( 40
) and v 0 (x) = 1 2π ∞ -∞ ṽ0 (k)e ikx dk (41)
and taking into account that

L 1 2 e ikx = ω(k)e ikx (42)
with

f (L 1 2 t)e ikx = f (ω(k)t)e ikx (43)
for any sufficiently smooth function f (ωt) = ∞ m=0 a m ω m t m and where ω(k) denotes the (positive) square-root of dispersion relation [START_REF] Gel'fand | Properties and Operations[END_REF]. It is now convenient to rewrite (38) in rather trivial manner

u(x, t) = cos (L 1 2 t) ∞ -∞ δ(x -ξ) u 0 (ξ)dξ + L -1 2 sin (L 1 2 t) ∞ -∞ δ(x -ξ) v 0 (ξ)dξ (44)
where the operators act only on the dependence on x. By using the Fourier representation of the δ-function [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] we have

Q(x, t) = L -1 2 sin (L 1 2 t)δ(x) = 1 2π ∞ -∞ e ikx sin (ω(k)t) ω(k) dk ( 45
)
and

∂ ∂t Q(x, t) = cos (L 1 2 t)δ(x) = 1 2π ∞ -∞ e ikx cos (ω(k)t)dk (46)
We can then write (44) in terms of the kernels Q(x, t) and

∂ ∂t Q(x, t) namely u(x, t) = ∞ -∞ ∂ ∂t Q(x -ξ, t)u 0 (ξ)dξ + ∞ -∞ Q(x -ξ, t)v 0 (ξ)dξ (47)
and the velocity field is written as

v(x, t) = ∞ -∞ ∂ 2 ∂t 2 Q(x -ξ, t)u 0 (ξ)dξ + ∞ -∞ ∂ ∂t Q(x -ξ, t)v 0 (ξ)dξ (48)
From relations (45) and ( 46) we observe the initial conditions

Q(x, t = 0) = 0 , ∂ ∂t Q(x, t = 0) = δ(x) , ∂ 2 ∂t 2 Q(x, t = 0) = 0 (49)
which guarantee that (47), (48) indeed solve the Cauchy problem. The problem is solved for our medium by the explicit determination of ( 45) and (46). To this end we evaluate relations (45) and ( 46) which can be written as8

Q(x, t) = ∞ n=0 (-1) n t 2n+1 (2n + 1)! q n (x) (50)
and

∂ ∂t Q(x, t) = ∞ n=0 (-1) n t 2n (2n)! q n (x) (51)
with the functions q n (x) defined by

q n (x) = L n δ(x) = 1 2π ∞ -∞ e ikx ω 2n (k) dk =: lim ǫ→0+ 1 π ∞ 0 e -kǫ ω 2n (k) cos kx dk , n = 0, 1, 2, .. ∈ N 0 (52)
The integral ∞ 0 e -kǫ k α cos kx dk exists for α > -1 and contains the entire admissible range 0 < δ < 2 of the Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF]. Hence we have

q n (x) = lim ǫ→0+ A n δ π ∞ 0 e -kǫ k δn cos kx dk = A n δ π Re(J nδ ) , n = 0, 1, 2, .. ∈ N 0 , 0 < δ < 2 (53)
with

J nδ = lim ǫ→0+ 1 (ǫ -i|x|) (nδ+1) ∞ 0 e -s s nδ ds = lim ǫ→0+ (nδ)! (ǫ -i|x|) (nδ+1)                i e i πδn 2 (nδ)! |x| nδ+1 , x = 0 , n = 1, 2, .. ∈ N lim ǫ→0+ 1 ǫ -i|x| = πδ(x) + iP( 1 x
) , n = 0 (54) where (nδ)! = Γ(nδ + 1) and P(..) denotes the principal value of (..). We obtain then

q n (x) = q n (|x|) =      A n δ π Re(J nδ ) = - A n δ π sin ( πδn 2 ) (nδ)! |x| nδ+1 , x = 0 , n = 1, 2, .. ∈ N , 0 < δ < 2 q 0 (x) = 1 π Re(J 0 ) = δ(x) = 0 , x = 0 (55)
where A δ is determined in (24). On the right-hand sides of (54), (55) we skipped a singular term which also occurs in the case n > 0 at x = 0. Therefore this relation excludes x = 0 for n > 0. We come back to this important point in section 6 by considering integrals of the J α for the entire range of their existence (α > -1).

The series (50), (51) together with functions (55) constitute the solution of the Cauchy problem being defined in the entire range 0 < δ < 2 of existence of Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF]. It is shown in appendix 8.2 that the series for (50), (51) converge uniformly absolutely for all x and t.

Retarded time domain Green's function

In view of the results of the last section it is only a small step to determine the retarded (or causal) dynamic Green's function. The dynamic Green's function is defined as the solution of the self-similar wave-equation

( ∂ ∂t + ǫ) 2 -∆ (δ,h,ζ) g(x, t) = δ(x)δ(t) (56)
with the self-similar Laplacian ∆ (δ,h,ζ) defined by [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF]. Causality means that g(x, t) must be zero before and nonzero only after the impact of the pulse δ(x)δ(t) i.e. g(x, t < 0) = 0 and g(x, t > 0) = 0 [START_REF] Jackson | Classical Electrodynamics, Third Edition[END_REF][START_REF] Michelitsch | Dynamic Eshelby tensor and potentials for ellipsoidal inclusions[END_REF]. We introduced in (56) an infinitesimal positive damping term ǫ → 0+ which breaks the time-inversion symmetry of the wave operator and guarantees in this way g(x, t) being the causal solution of (56). The physical interpretation of g is the displacement field due to a δ-type force density of the form of the right hand side of (56). The Green's function defined by (56) solves the dynamic problem

( ∂ ∂t + ǫ) 2 -∆ (δ,h,ζ) u(x, t) = f (x, t) (57)
where f (x, t) is the density of external forces and u(x, t) the corresponding displacement field by the convolution

u(x, t) = ∞ -∞ t -∞ g(x -ξ, t -τ )f (ξ, τ ) dξ dτ ( 58
)
where g(x, t) indicates the Green's function defined by (56). The upper limit in the time-integration in ( 58) is due to the causality of g(x, t). With the Fourier-transformation

g(x, t) = 1 2π ∞ -∞ g(k, t)e ikx dk (59)
and

δ(x) = 1 2π ∞ -∞ e ikx dk (60)
yields (56) the evolution equation for the normal amplitudes g(k, t) in the form

( ∂ ∂t + ǫ) 2 + ω 2 (k) g(k, t) = δ(t) (61)
which is nothing but the equation for the causal Green's function of a damped harmonic oscillator with damping constant 9 ǫ > 0 and eigenfrequency ω(k) which has the (causal) solution of the well-known form 10

g(k, t) = e -ǫt Θ(t) sin ω(k)t ω(k) (62)
where Θ(t) denotes the Heaviside unit-step function being defined as Θ(t) = 1 for t > 0 and Θ(t) = 0 for t < 0. (62) holds for any (also non-infinitesimal) damping ǫ > 0 where the dissipative factor e -ǫt can be skipped in the limit of an infinitesimal ǫ > 0. The frequency

ω(k) = √ A δ |k| δ 2
denotes the positive root of the dispersion relation [START_REF] Gel'fand | Properties and Operations[END_REF] holding for 0 < δ < 2. In view of the relation

L -1 2 sin L 1 2 t e ikx = sin ω(k)t ω(k) e ikx (63)
with L = -∆ δ,h,ζ we have the retarded space-time Green's function g(x, t) in the form

g(x, t) = e -ǫt Θ(t)L -1 2 sin (L 1 2 t) δ(x) = e -ǫt Θ(t) Q(x, t) (64)
9 ǫ → 0+ being positive and infinitesimal being sufficient to obtain the causal solution. 10 See any textbook of theoretical physics or e.g. [START_REF] Michelitsch | Dynamic Eshelby tensor and potentials for ellipsoidal inclusions[END_REF].

where Q(x, t) has been evaluated in relations (50) ff. Some remarks on the dynamic Green's function in the space-frequency domain For the sake of completeness we give a brief representation of the frequency-space Green's function, i.e. Green's function of the self-similar Helmholtz equation. The Helmholtz Green's function is defined as

ĝ(x, ω) = ∞ 0 e iωt g(x, t) dt (65)
where g(x, t) is the retarded space-time-domain Green's function (64) and the lower integration limit is due to causality. (65) solves the self-similar Helmholtz-equation

(ω + iǫ) 2 + ∆ (δ,h,ζ) ĝ(x, ω) = -δ(x) (66)
where ǫ > 0 is a positive infinitesimal damping constant which guarantees existence of the Helmholtz Green's operator defined in (67) below and causality in the time domain. We can invert this Helmholtz equation with ∆ (δ,h,ζ) = -L where the Helmholtz Green's function can be written as

ĝ(x, ω) = L -(ω + iǫ) 2 -1 δ(x) (67)
with the Green's operator (resolvent operator) of the frequency-domain L -(ω + iǫ) 2 -1 .

Self-similar diffusion problem -Lévi flights

This section is devoted to the analysis of the diffusion problem governed by the self-similar Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF].

We consider an ensemble of particles with local concentration ρ(x, t). We also can conceive this problem to be defined below as a fractional differential equation for a (probability)-distribution ρ(x, t). The idea to represent statistical distributions as solutions of fractional differential equations is actually not new, we refer for instance to the paper of Li et al. [START_REF] Li | Fractional Differential Equations and Stable Distributions[END_REF] where "stable" distributions are generated by fractional differential equations. However, so far the fractional differential equations generating statistical distributions as solutions were rather a kind of "guesswork" without having an intuitive physical "picture" about the generating fractional differential equation. The goal of this section is to analyze a diffusion-equation employing the self-similar Laplacian (7) of the form

∂ ∂t ρ(x, t) = -Lρ(x, t) (68)
where -L = ∆ (δ,h,ǫ) indicates the self-similar Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF]. We consider this problem in the domain -∞ < x < ∞ and t > 0. It is important that we restrict on t > 0 as diffussion is a time-irreversible process 11which is expressed by the odd-order of the time derivative in (68). ρ(x, t) denotes the density of the diffusing particles but can be also conceived as a probability-density distribution-function. (68) recovers the traditional diffusion equation by replacing the self-similar Laplacian ∆ δ,h,ζ by the traditional 1D-Laplacian ∂ 2 ∂x 2 leading to a gaussian distribution.

We give an intuitive physical picture of the diffusion processes described by (68) as follows: Let us assume we have ρ(x, t)dx particles at time t at location x. Any particle is allowed to jump at time t from position x to arrive at t + dt at any location x ± τ different from x (with 0 < τ < ∞). The jump-rate of this process is measured by (ρ(x + τ, t) + ρ(x -τ, t) -2ρ(x, t))/τ -(1+δ) which is given by the integrand of (7) where 0 < δ < 2. We observe the following: The jump-rate from x to x ± τ is zero if (ρ(x + τ, t) + ρ(x -τ, t) -2ρ(x, t)) = 0, i.e. there are no jumps for a equal distribution ρ = const. Jumps to "far" locations are more seldom as jumps to "close" locations for equal (ρ(x + τ, t) + ρ(x -τ, t) -2ρ(x, t)) = 0. The possibility of a particle to make jumps of arbitrary distances is expressed by the non-locality and continuity of Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF]. Particle-jumps to any location are possible in (68), whereas non-local jumps are suppressed in the traditional (gaussian) case which is reflected by the locality of the traditional Laplacian ∂ 2 ∂x 2 . The non-local Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF] sums up over all possible jump-events by which particles can reach to or escape from a space-point x. For ρ = const the jumping rate is zero (since L1 = 0) and, as we will show, the equal-distribution is a stationary solution and ρ = 0 an attractor for t → ∞ for any initial distribution ρ 0 (x) just as in the case of gaussian distribution.

If we assume ρ(x, t) to be the probability distribution of a single particle and ρ(x, t)dx to be the probability to find a particle in the interval within [x, x + dx] at time t, then the trajectory of the particle propagating due to probability-distribution ρ(x, t) which evolves according to (68) is discontinuous and erratic whereas in the gaussian case the particle-motion describes a continuous trajectory. The discontinuous characteristics of the particle trajectory associated with (68) becomes more and more pronounced as δ → 0 which is due to the weaker decay of the kernel τ -(δ+1) for τ → ∞. We should like to mention that motions characterized by jump-probability distributions of the form of a power law as τ -(δ+1) are known in the literature as Lévi-flights where δ is called Lévi-parameter. Lévi flights correspond to Lévi-stable probability distributions for 0 < δ ≤ 2 [START_REF] Mandelbrot | Les objets fractals -form hasard et dimension[END_REF]. Typical features of Lévi-distributions are their infinite variances in the range 0 < δ < 2. The present analysis will end up in probability-distributions of exactly this type.

In order to define diffusion problem (68) as initial value problem, we prescribe an initial distribution ρ 0 (x) at t = 0 as

ρ(x, t = 0) = ρ 0 (x) (69)
We assume that the particle number is conserved in time, there is neither particle generation nor annihilation. The diffusion equation (68) describes then the continuity equation accounting for the particle (probability) balance at space-point x and time t having the form

∂ ∂t ρ(x, t) + ∇ • j(x, t) = 0 ( 70
)
where ∇ = ∂ ∂x denotes the usual 1D gradient operator. Unlike in the Gaussian case where the traditional Laplacian comes into play by assuming Fick's law as a consequence of a gradient dynamics as constitutive law, in our case the constitutive law is instead of a (local) gradient a non-local constitutive law for the particle (probability) flux. An explicit form of the particle (probability) flux is obtained in view of (8) as

j(x, t) = - h δ ζδ ∞ 0 (ρ(x + τ, t) -ρ(x -τ, t)) τ δ dτ , 0 < δ < 2 (71)
which is our self-similar constitutive law and can be conceived as the self-similar analogue to Fick's law j gauss (x, t) = -∇ρ(x, t) of the Gaussian case. In (71) an irrelevant integration-constant has been neglected. We can consider (71) as definition of the self-adjoint part of the self-similar gradient operator acting on ρ(x, t).

The distribution function ρ(x, t) is required to fulfil the normalization condition

∞ -∞ ρ(x, t)dx = ∞ -∞ ρ 0 (x)dx = 1 , ∀t > 0 (72)
(72) indicates that ρ(x, t) has to be a normalized distribution. ρ(x, t) is uniquely determined by ( 68) and (69) and can be written in the form

ρ(x, t) = e -Lt ρ 0 (x) (73)
where we restrict us to t > 0. The question to be answered in the following is: Does (73) represent a normalized distribution, i.e. fulfills (72) under the condition that ρ 0 (x) represents a normalized distribution? We can answer this question by "yes" by giving the following formal prove by integrating (73)

∞ -∞ ρ(x, t) dx = e -Lt ∞ -∞ ρ 0 (x) dx = e -Lt 1 = 1 ( 74
)
where we take into account that L n 1 = δ n0 since Laplacian (7) applied to a constant yields zero. Let evaluate (73) and deduce the propagator (kernel) that represents ρ(x, t) in terms of a convolution

ρ(x, t) = ∞ -∞ W (x -ξ, t)ρ 0 (ξ)dξ = e -Lt ∞ -∞ δ(x -ξ)ρ 0 (ξ) dξ (75)
The propagator W (x, t) is obtained by

W (x, t) = e -Lt δ(x) (76)
and fulfils the initial condition

W (x, t = 0) = δ(x) (77)
The necessary and sufficient condition that W (x, t) describes a normalized probability distribution is

∞ -∞ W (x, t) dx = 1 (78)
which is indeed fulfilled as can be seen by putting ρ 0 (x) = δ(x) in (74). If ρ(x, t) is conceived as a probabilitydistribution W (x, t) is the conditional probability density to find a particle which was at t = 0 located at x = 0 at time t > 0 at point x. The propagator W (x, t) can be evaluated by using Le ikx = ω 2 (k)e ikx in the form

W (x, t) = e -Lt δ(x) = 1 2π ∞ -∞ e ikx e -ω 2 (k)t dk = 1 π ∞ 0 e -A δ k δ t cos (kx) dk ( 79
)
where ω 2 (k) = A|k| δ is the dispersion relation [START_REF] Gel'fand | Properties and Operations[END_REF] which is defined in 0 < δ < 2. Integral (79) exists in the whole range of 0 < δ < 2 where A δ > 0 for t > 0. We observe further because of ω 2 (k) ≥ 0 that e -ω 2 (k)t → 0 at t → ∞ thus (79) tends to zero as t tends to infinity lim t→∞ W (x, t) = 0 (80) so the stationary solution ρ ∞ = 0 is always an attractor independent of the initial distribution ρ 0 (x) in the physical admissible interval 0 < δ < 2. It is important to note that for any finite 0 < t < ∞ (80) maintains in some regions non-zero (positive) values in order to fulfil (78). The goal is now to evaluate (79).

It can be further seen in ( 79) since e -A δ k δ t is a monotonous decreasing function in k that the positive contributions due to cos kx dominate the negative contributions in the last integral of (79) so that W (x, t) fulfills another necessary characteristics of a probability-distribution, namely W (x, t) ≥ 0 ∀x ∈ R and t > 0 (81)

So we can infer that the initial value problem defined by ( 68), (69) with the self-similar Laplacian (7) indeed describes a diffusion problem (transport problem) under the condition that (69) is a normalized distribution. It remains us now to evaluate (79) in more explicit form

W (x, t) = 1 π Re ∞ 0 e ikx-A δ k δ t dk , 0 < δ < 2 (82)
where 0 < δ < 2 is the range of existence of A δ . Distributions of the form (79), (82) have already been proposed by Lévi [START_REF] Mandelbrot | Fractales, hasard et finance[END_REF][START_REF] Lévi | Processus stochastiques et mouvement Brownien[END_REF] and refers to the category of a "Lévi-stable" or "L-stable" distribution (for further details on L-stable distribution we refer e.g. to [START_REF] Mandelbrot | Fractales, hasard et finance[END_REF], p. 106ff. and the references therein). For δ = 1 we can directly evaluate (82) directly and obtain

W δ=1 (x, t) = Re 1 π ∞ 0 e -k(A 1 t-ix) dk ( 83
)
which yields

W δ=1 (x, t) = 1 π A 1 t (x 2 + (A 1 t) 2 ) ( 84
)
where A 1 = hπ ζ . Relation (84) represents spatially a distribution of the Cauchy type with A 1 t being a "parameter". One verifies directly that (84) fulfills the conditions (77), ( 78) and (80). Lévi already gave static spatial probability-distributions of the general form (82) [START_REF] Mandelbrot | Les objets fractals -form hasard et dimension[END_REF] and found that in general they are admissible for 0 < δ ≤ 2 where δ = 1 corresponds to the Cauchy-distribution and δ = 2 to the Gaussian distribution. For δ = 2 the Laplcian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF] does not exist and hence A 2 is not defined in our approach.

Mandelbrot showed that L-stable distributions play a crucial role in the description of economic processes and stock courses and describe the statistics of fractal irregular trajectories. The irregularity of these trajectories decreases from δ = 0 to δ → 2 [START_REF] Mandelbrot | Les objets fractals -form hasard et dimension[END_REF] which is consistent with our above given physical picture. The irregularity is due to the fact that (82) produces in the admissible range of existence of the Laplacian (7) 0 < δ < 2 infinite fluctuations due to discontinuous trajectories. The Cauchy distribution (δ = 1) itself refers also to this category. Only the Gaussian case corresponds to a diffusion equation with a traditional Laplacian describing continuous trajectories (Brownian motion) with finite mean fluctuation (variance). A consequence is that the unjustified use of Gaussian statistics underestimates fluctuations and risks! We refer in this context to [START_REF] Mandelbrot | The Fractal Geometry of Nature[END_REF][START_REF] Mandelbrot | Fractales, hasard et finance[END_REF][START_REF] Mandelbrot | Les objets fractals -form hasard et dimension[END_REF][START_REF]Universalités et Fractales[END_REF] and the references therein. We will reconsider briefly this important point at the end of this section.

It still remains to evaluate (82) which we do by expanding e -ω 2 (k)t in a series and arrive at the expression

W (x, t) = ∞ n=0 (-1) n t n n! q n (x) , 0 < δ < 1 ( 85
)
where this series converges only in the range 0 < δ < 1 which is shown in appendix 8.3. In (85) appear the functions

q n (x) = L n δ(x) = 1 2π ∞ -∞ e ikx ω 2n (k) dk ( 86
)
which where already determined in the last section by ( 52)-( 55). Due to their importance for our approach we devote the next section 6 to their thorough analysis. The series (85) writes

W (x, t) = 1 π lim ǫ→0+ Re ∞ n=0 (-1) n ((δn)!(A δ t) n ) n!(ǫ -i|x|) nδ+1 = δ(x) + 1 π lim ǫ→0+ Re ∞ n=1 (-1) n ((δn)!)(A δ t) n n!(ǫ -i|x|) nδ+1 (87)
which assumes for x = 0

W (x, t) = 1 π ∞ n=1 (-1) n-1 (nδ)! n! sin ( πnδ 2 ) A n δ t n |x| nδ+1 , 0 < δ < 1 ( 88
)
where the singular parts at x = 0 vanishing for x = 0 have been omitted. (88) has the form

W (x, t) = - 1 π|x| Im{w(ξ(x, t))} (89)
when we introduce the variable

ξ(x, t) = t |x| δ e πiδ 2 A δ (90)
with the function w(ξ)

w(ξ) = ∞ n=1 (-1) n (nδ)! n! ξ n , 0 < δ < 1 ( 91
)
where series representations (85), (87) converge ∀ξ in the interval 0 < δ < 1 (appendix 8.3). Let us now discuss some of the main characteristics of the distribution W (x, t) which is defined in the whole range 0 < δ < 2: In view of (79) and (55) we observe that W (x, t) = W (-x, t) = W (|x|, t) is a symmetric distribution with respect to x and as a consequence all moments of odd order

< x 2n+1 >= ∞ -∞ W (x, t) x 2n+1 dx = 0 , n = 0, 1, .. ∈ N 0 (92)
are vanishing especially the mean-value < x >= 0. The moments of even order are obtained from

< x 2n >= ∞ -∞ W (x, t) x 2n dx → ∞ , n = 1, .. ∈ N (93)
are diverging. We can conclude without any further calculation that (93) is divergent since any real valued power function (except a constant) is not admissible in [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF]. The existence of (7) however is achieved for certain complex valued power functions as we show in the next section 6.

Self-similar potentials

Finally we analyze more closely functions of the type q n (x) of (86) which were determined in (52)-(55). To this end let us evoke the definition of the q n (x) = L n δ(x) which was defined by using the Laplacian ( 7), [START_REF] Bondarenko | Self Similar Spectrum of Fractal Lattice[END_REF]. We should emphasize that ∆ (δ,h,ζ) = -L depends on the continuous parameter δ which was restricted in [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF], [START_REF] Bondarenko | Self Similar Spectrum of Fractal Lattice[END_REF] to the interval 0 < δ < 2 where the upper limit δ < 2 came into play since 2u(x)

-u(x + τ ) -u(x -τ ) → C(x)τ 2
is a quadratic function of τ for small τ . In this subsection we will show that for a certain class of singular functions, namely functions of the form b nδ (x) = L n δ(x)/A n δ , the upper limit δ < 2 does not exist and for which the condition δ > 0 is sufficient. We call these functions due to their importance as solutions of the Laplaceequation ( 7) as "self-similar potentials". Their singular behavior at x = 0 becomes especially important when we consider integrals or convolutions of these functions. Whereas any real-valued power-functions x β are not admissible in [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF]. Self-similar potentials to be analyzed, however, are admissible and give the only way to an admissible definition of powers L n . Let us consider the Fourier transformation of |k| α e -|k|ǫ with ǫ > 0, namely

12 b α (x) = lim ǫ→0+ 1 2π ∞ -∞ e ikx-|k|ǫ |k| α dk = lim ǫ→0+ 1 π ∞ 0 e -ǫk k α cos kx dk , α > -1 ∈ R (94)
which exists for α > -1. This function can be identified with the q n (x) of (86) for α = nδ. We can rewrite (94) by introducing the integration variable s = k(ǫ -i|x|) in the form

b α (x) = Re lim ǫ→0+ 1 π 1 (ǫ -i|x|) α+1 ∞ 0 e -s s α ds = α! π Re lim ǫ→0+ i α+1 (x + iǫ) α+1 , α > -1 ∈ R (95)
which holds for α > -1 and where α! = Γ(α + 1). We note that (x + iǫ) -α-1 assumes the complex conjugate value when replacing x ↔ -x. So its real-part depends only on |x| and we can rewrite

b α (x) = b α (|x|) = α! π Re lim ǫ→0+ i α+1 (|x| + iǫ) α+1 , α > -1 ∈ R (96)
Expressions (95), (96) are in accordance with with the result given by Gel'fand and Shilov for the Fourier transform of |k| α (page 447, equation ( 13) in [START_REF] Gel'fand | Properties and Operations[END_REF]).

We summarize the following important observations:

(i) b α (x) = b α (-x) = b α (|x|) is a symmetric function in x for all α > -1.
(ii) b α=0 (x) = δ(x) represents the usual Dirac-δ-function, i.e. is localized at x = 0.

(iii) For α = 2n with n = 0, 1, 2, .. ∈ N 0 (95) takes the form of even derivatives of the δ-function b 2n (x) = (-1) n d 2n dx 2n δ(x) (97) 12 For further details on the Fourier transform of |k| α , see [START_REF] Gel'fand | Properties and Operations[END_REF] which are localized at x = 0.

(iv) α > -1 ∈ R: For x = 0 (95), b α has the explicit form (for x = 0 we can put directly ǫ = 0 in (95))

b α (x) = - α! π|x| α+1 sin ( απ 2 ) (98)
where we directly verify that this expression is zero for α = 2n (in accordance with case (iii) for x = 0). For α > -1 ∈ R / ∈ 0, 2, 4, .. expression (98) is non-zero for x = 0 and hence in contrast to the Dirac's δ-function, non-local. Expression (98) includes also the case of odd integers α = α n = 2n + 1 where n = 0, 1, 2, .. ∈ N 0 .

We observe that b α (x) defined by (94) can be conceived as the Fourier transform of |k| α which exists only for α > -1. For exponents α < -1 the integral (94) diverges. However, we can define a regularized Fourier transform in the spirit of generalized functions as given in Gel'fand and Shilov [START_REF] Gel'fand | Properties and Operations[END_REF]) which exists also for α < -1. We devote appendix 8.4 to this case.

Let us consider now the following integral

∞ -∞ b α (x)dx = ∞ -∞ e -|k|ǫ |k| α δ(k)dk =    0 , α > 0 1 , α = 0 ∞ , -1 < α < 0 (99) (v)
For α > 0 we observe that integral (99) is vanishing. On the other hand we observe that b α (x) = 0 for x = 0 and sign(b α (x)) = -sign(sin ( απ 2 )) is the the same ∀x = 0. Let us raise the following question: How is it possible that the integral (99) is vanishing for α > 0 whereas b α = 0 having the same sign as -sign(sin ( απ 2 )) almost everywhere, i.e. ∀x = 0? To give the answer, let us consider the integral is diverging for α > 0 as a -α ! That means if we imagine for a moment |b α | being a mass density, that the total mass concentrated in the interval 0 ≤ x ≤ a is the same as the total mass in the interval a < x < ∞. We conclude that b α (x) has, in dependence on α, several oscillatoric peaks of alternate signs localized for ǫ → 0+ infinitely close to x = 0. The integral over these localized oscillations up to a finite limit a > 0 compensates integral (100). The strength of these oscillatoric peaks are "infinitely stronger" as that one of a δ-function.

I α (a) = ∞ a>0 b α (x)dx = - (α -1)! πa α sin ( πα
The closed form expression of the b α which takes into account this singularity is therefore (95) which fulfils for instance the "desired" property of a fractional derivative of order α > 0 (e.g. [START_REF] Jumarie | From Self-Similarity to Fractional Derivative of non-differentiable functions via Mittag-Leffler Function[END_REF]) that it yields zero when applied to a constant. This property requires that

a 0 f (x)dx = Const × a -α (106)
which is obviously not solved for α > 0 by any real valued power-function f (x) = const x -α-1 ∈ R since its primitive in (106) diverges at x = 0. In contrast b α (x) solves ( 106)

Re ∞ 0 dx (ǫ -ix) α+1 = Re (-i) α 1 (ǫ -ix) α | ∞ 0 = 0 , α > 0 ( 107
)
for any ǫ > 0 since the lower limit of the integral is purely imaginary and the upper limit vanishing for α > 0. Hence (95) defines for α > 0 an integration kernel for a fractional derivative which yields zero when applied to a constant which is due to the vanishing of (99) which we can indeed verify directly. We analyze in appendix 8.1 further properties of b α .

Let us now return to the functions q n (x) of ( 86): Taking into account (95) have with α = nδ > -1 and with

b nδ (x) = L n A n δ δ(x) = (nδ)! π lim ǫ→0+ Re i nδ+1 (|x| + iǫ) nδ+1 , nδ > -1 (108)
where A δ is given in (24). For n = 1 we obtain

b δ (x) = 1 A δ Lδ(x) = δ! π lim ǫ→0+ Re i δ+1 (|x| + iǫ) δ+1 (109)
being in accordance with q 1 (x) = Lδ(x) = -∆ (δ,h,ζ) δ(x) obtained by evaluation of ( 7) or ( 8) for x = 0. From (108) one can recover the static Green's function by ( 13)

q -1 (x) = b -δ (x) A δ = L -1 δ(x) = (-δ)! πA δ Re i 1-δ (|x| + iǫ) 1-δ , 0 < δ < 2 (110)
which indeed recovers expression (27) for this Green's function q -1 (x) = g(x). (110) takes into account in a "correct" way the singular behavior at x = 0 and is therefore more general as (27) holding only for x = 0. Whereas the b nδ (x) hold for nδ > -1, the functions q n (x) = A n δ b nδ (x) are only physically meaningful where A δ > 0 and where (7) exists namely 0 < δ < 2. The range of validity of the b α (x) can be mathematically further extended to all α < -1 if one utilizes instead of [START_REF] Ross | Functions that have no First Order Derivative might have Fractional Derivatives of all Orders Less than One[END_REF] the regularized definition of the Γ-function (34).

The performed analysis gives reason to the following observation: There is a general link between noninteger α including odd integers α and the non-locality and self-similarity of physical properties. This behavior is also reflected by the non-locality of our Laplacian [START_REF] Kigami | A harmonic calculus on the Sierpinski spaces[END_REF]. Intuitively in a wider sense one is tempted to state: even integer-exponents α (case (iii)) correspond to local ("regular"), non-integer and odd exponents α (case (iv)) to non-local, self-similar ("irregular")13 physical behavior.

Conclusions

We have analyzed an elastic medium with self-similar elastic energy-density. The source of the physics in this continuum is the self-similar Laplacian (7) having the form of a non-local convolution with a power-law convolution kernel. This Laplacian can be represented in terms of fractional derivatives and is obtained by the continuous limiting case of the discrete self-similar Laplacian introduced in [START_REF] Michelitsch | Dispersion relations and wave operators in self-similar quasicontinuous linear chains[END_REF]. The resulting equations of motion are self-adjoint partial fractional differential equations which are only in the range 0 < δ < 2 physically meaningful.

We deduced closed form solutions (i) for the static Green's function (displacement field due to a unit δ-force), (ii) for the Cauchy problem (two convolution kernels "propagators"), (iii) for the solution of a selfsimilar diffusion problem. The solution was obtained in terms of a convolution kernel solving the initial value problem. It has been found that the obtained convolution kernel represents normalized Lévi-stable probablity distributions with inifinite variances (mean fluctuations) in their range of definition 0 < δ < 2.

As a spin-off result we obtained a set of functions b nδ (x) which we denoted as self-similar potentials appearing as regularized Fourier transforms of |k| α . These functions are to be conceived as generalized functions in the distributional sense of Gel'fand and Shilov [START_REF] Gel'fand | Properties and Operations[END_REF]. Self-similar potentials are special solutions of the self-similar Poisson's equation and play in statics the analogous role as Newtonian potentials for the traditional Poisson equation.

It has been demonstrated that the concept of self-similar functions introduced in [START_REF] Michelitsch | Dispersion relations and wave operators in self-similar quasicontinuous linear chains[END_REF] is useful to describe material systems with self-similar interactions. Our approach opens the door to a continuum description of a rich fond of new physical especially dynamic phenomena in such systems. Especially interesting are the propability-distributions obtained as solutions of the self-similar (fractional) diffusion equation (68). These probability-distributions are Lévi-stable which are known to be a characteristic feature of the statistics of irregular and erratic motions characterized by discontinuous trajectories (Lévi-flights) [START_REF] Mandelbrot | The Fractal Geometry of Nature[END_REF][START_REF] Mandelbrot | Les objets fractals -form hasard et dimension[END_REF]. It is mentionworthy that our diffussion model recovers the range 0 < δ < 2 of admissible Lévi parameter δ given previously in the literature (e.g. [START_REF] Mandelbrot | The Fractal Geometry of Nature[END_REF][START_REF] Mandelbrot | Les objets fractals -form hasard et dimension[END_REF]).

We have further formulated the 1D continuity equation for the self-similar diffusion problem and formulated the self-similar probability-flux (Eq. ( 71)). An interesting open point which can be treated by the present approach remains the formulation of the non-local self-similar Gauss-theorem in dimensions higher than one, and more general to formulate continuity equations and related problems in self-similar continua embedded into the 2D or 3D space.

The present approach can be a point of departure to answer such questions in a rather rigorous way. For example the present approach can be considered as a point of departure to any general self-similar field theory such as continuum mechanics and electrodynamics. We hope the present paper stimulates work in such directions.

Appendix

Fractional derivative

In view of the analysis of the last section we can easily recall a fractional derivative D α x f (x) which fulfills for α > 0 all desirable properties of a derivative. We call positive orders (fractional) derivatives and negative orders (fractional) integrals. Let us consider the case α > -1 which can however be extended to α < -1 in the spirit of generalized functions [START_REF] Gel'fand | Properties and Operations[END_REF] as outlined in appendix 8.4.

Let us first of all determine

y α (x) = D α x δ(x) , α > -1 (111)
where we call y α (x) a fractional derivative. For α = 0 the solution of should be y α=0 (x) = δ(x). From (111) follows then for the definition of this fractional derivative applied on a function f (x) the convolution

D α x f (x) = ∞ -∞ y α (x -τ )f (τ )dτ , α > -1 (112)
with the kernel y α (x) (115) to be determined. Further we assume that D α x e ikx = (ik) α e ikx so we can write y α (x) write as a Fourier integral ) where P is given by

P (ξ) = ∞ n=1 (nδ)! (2n)! ξ n , 0 < δ < 2 (119)
In order to judge the convergence of this series the asymptotic representation of (nδ)! (2n)! for n >> 1 will be useful. To this end we put s(n) = 2n(1 -δ 2 )) with nδ = 2n -s (nδ)! (2n

)! = (2n -s(n))! 2n! → 1 (2n) 2n(1-δ 2 )
, 0 < δ < 2 (120)

From (120) we can determine the radius of convergence ρ c of the series (119) where the series ∞ n 0 a n ξ n converges absolutely for |ξ| < ρ c with

ρ c = lim n→∞ | a n a n+1 | = lim n→∞ (2ne) 1-δ 2 = ∞ , 0 < δ < 2 (121)
since 1 -δ 2 > 0. Hence series (119) converges for all ξ absolutely which guarantees the convergence of (50) and (51) for all x and t for δ being in the interval 0 < δ < 2.

Convergence of series (85)

We consider the convergence of ( 85 The radius of convergence r c of this series is

r c = lim n→∞ (nδ)! n!
(n + 1)! ((n + 1)δ)! = (n + 1) (nδ)! ((n + 1)δ)!

→ n 1-δ δ -δ → ∞ , 0 < δ < 1 (125) Hence (124) converges for all ξ absolutely uniformly. Hence (85) converges in the entire domain -∞ < x < ∞ and t > 0 for δ being in the interval 0 < δ < 1.

8.4 Some remarks on the Fourier-integral (94) for α < -1 A further observation is important in b α (x) of (94) which is the Fourier transform of |k| α exists for α > -1. For exponents α < -1 the integral (94) diverges. However, we can define a Fourier transform of Re{(-i) α (|k| + iǫ) α } in the limiting case ǫ → 0+ which exists also for α < -1.

We generalize the definition of b α (x) (94) for α < -1 by replacing |k| α → 1

where Γ(-α) is for α < -1 well defined in [START_REF] Ross | Functions that have no First Order Derivative might have Fractional Derivatives of all Orders Less than One[END_REF].

which has the same form as expression (98) which holds for α > -1 but with the generalized definition of α! given in (34). In conclusion (131) holds for α ∈ R / ∈ ±1, ±3, .. ± (2n + 1), ... For cases α being even integers, expression (131) is not valid, we refer in this context further to [START_REF] Gel'fand | Properties and Operations[END_REF] where the Fourier transforms 1/|k| 2n+1 (n = 0, 1, 2..) are further analyzed.

2 ∞-eJ

 2 the property lim a→0+ |I α (a)| → ∞. Let us again ask, why does (100) not converge in the limiting case a → 0+ towards zero for α > 0 as indicated in (99) since b α (x) = b α (-x) is an even function? If integral (99) indeed is vanishing, then (100) should be compensated by the integralJ α (a) = α (x)dx + ∞ a b α (x)dx = 1 ∞ b α (x)dx = 0 (b(x) = b(|x|)).To evaluate this integral we have to use (94) or directly (96) which hold ∀x including x = 0 to arrive atJ α (a) = lim ǫ→0+ -ǫk k α-1 sin (|k|a) dk = (α -1)! π Im 1 (ǫ -ia) α(103)since a > 0 being non-zero we can further evaluate (103) by putting ǫ = 0 to arrive at J α (a) = (100) so that J α (a) + I α (a) = 0 thus (99) indeed is fulfilled with the remarkable property lim a→0+

e 1 )

 1 -ǫk k α e i(kx+ απ 2 ) + e -i(kx+ απ 2 ) x = 0 (115) assumes the formy α (x) = -α! π|x| α+1 sin πα(116)We verify that for integer α = 0, 1, 2, .. ∈ N 0 (116) is vanishing for x = 0 and (115) yieldsy n (x) = (-reproducing in the integer-case the correct localized the integer-order derivatives.

8. 2

 2 Convergence of series (50), (51) It still remains to verify the convergence of the series (50) and (51): To this end we consider series (51) which has the form ∂ ∂t Q(x, t) = δ(x)q 0 (x) = δ(x) and P is only a function of ξ = A δ t 2 e πiδ 2 |x| δ

cos πα 2 1 π cos πα 2 Re 3 ,

 223 Re(ǫ -i|k|) α ) in the form b α (x) = lim ǫ→0+ iǫ) α cos kx dk , α ∈ R / ∈ ±1, ±3, .. ± (2n + 1), .. (126) which exists α ∈ R where the zeros of the cosine have to be excluded. For α > -1 (126) is identical with (94) and for α < -1 it complements the representation (94) which diverges in this case. Let us especially evaluate (126) in the "forbidden" case α < -1: Taking into account (ǫ -i|k|) α = 1 Γ(-α) ∞ 0 e -τ (ǫ-i|k|) τ -α-1 dτ (127) existing for α < -1 we obtain the representation b α (x) = lim .. -(2n + 1), ..

  By taking into account the Euler relation (34) we arrive at b α (x) = -α! π|x| α+1 sin απ 2 , α < -1, / ∈ -1, -3, .. -(2n + 1), ..

Where that "point" is a fixed-point of that scaling operation.

The additional factor 1/2 in the elastic energy avoids double counting.

We have -1 = e iπ with (-1) -δ = e -πiδ as we use throughout this paper for any complex number z = |z|e iϕ the principal value -π < ϕ = Arg(z) ≤ π for its argument ϕ.

Re(..) denotes the real-and Im(..) the imaginary part of a complex number (..).

See details on integrals of this type in appendix 8.4.

This definition of α! maintains the representations of regularized integrals of the form (126) for all admissible values α ∈ R.

i.e. L = L +

Where the convergence of these series will be verified in appendix 8.2.

That means the time-inverse process is physically inadmissible.

The meaning of the term "irregular" is here used in the sense of "erratic", it does not mean the absence of rules.