Andrei P Kirilyuk
email: andrei.kirilyuk@gmail.com

The Exact Image of Unified Reality, from the Electron to Consciousness

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The problem: Increasingly pressing inefficiency of traditional mathematics

The popular Wigner's thesis of 1960 about the "unreasonable efficiency of mathematics in the natural sciences" becomes increasingly and even catastrophically compromised in the last decades of first stagnation and then deep crisis of fundamental science, sometimes referred to as the "end of science". While one may evoke various explanations of this obvious "saturation of scientific discoveries", it is definitely related to the mathematical framework of fundamental science (and thus mathematics in the whole), just due to the strong connection between the two, even irrespective of its underlying interpretation.

Contrary to what one might assume based on the huge technological, applied-science progress of the last time, this problem of "suddenly" growing inefficiency of traditional mathematical methods of fundamental science is far from only professional scientific importance. It's rather the opposite relation that becomes the more and more evident: those apparent successes of powerful, but only empirically developing technology meet quickly growing obstacles of both internal and external origin, which can be summarised as "complexity crisis" in either technological drivers or social development around them.

While modern problems become the more and more "critically complex" due to the "global", multicomponent interaction processes involved, the dominating exact-science tools remain basically as "simple" as before, oriented to smooth, effectively one-dimensional, P a g e | 2

"integrable" behaviour and structures. The civilisation development itself asks now for the new power of rigorously complete, reliable solution to urgent practical problems of the new level of complexity determining the entire further destiny of humanity.

Even the last-time tendency in fundamental science based on the peaking "magic" power of high-tech observation methods shows a strange "inverted-progress effect", where the number of difficult, "unsolvable" problems, or "mysteries", remarkably grows ("dark matter", "crisis in cosmology", etc.), thus compromising the externally bright picture of spectacular science progress of the twentieth century. The accumulated "old" mysteries of science (quantum mechanics, relativity, gravity, unification, high-temperature superconductivity, etc.) also become increasingly disturbing on this background, as all of it seems to push such kind of "objective" knowledge system rather in the opposite direction of supernatural and therefore irresolvable enigmas. While basic science sadly declines, superstitions massively flourish, in this strange age of apparently triumphant technology.

The persisting traditional "noncomputability" of the huge spectrum of non-exact, "subjective" knowledge, from "empirical" sciences to the humanities and spiritual matters, also reappears as an integral part of the same critical problem and "supernatural" mystery, especially because the catastrophically peaking interaction power of the now "globalised", practically unified world populations, beliefs and professions urgently needs the respective qualitatively more powerful and unified knowledge basis.

The outlined crisis in today's exact sciences and knowledge system in the whole suggests only one kind of "really good" solution: the new, explicitly extended and now intrinsically complete content of rigorous, mathematical description of reality, leaving no place for postulated supernatural mysteries, arbitrary guesses and impasses of development. In the following sections we outline major features of such new mathematical basis for the causally complete and intrinsically sustainable scientific knowledge, in the form of extended unified mathematics of unreduced dynamic complexity (see [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Universal Symmetry of Complexity and Its Manifestations at Different Levels of World Dynamics[END_REF][START_REF] Kirilyuk | Dynamically Multivalued Self-Organisation and Probabilistic Structure Formation Processes[END_REF][START_REF] Kirilyuk | Complex-Dynamical Extension of the Fractal Paradigm and Its Applications in Life Sciences[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF] for detailed technical accounts).

The solution:

Extended unified mathematics of the unreduced dynamic complexity (1) What may be missing. While looking for the causally complete mathematical framework as the solution to the problem of growing inefficiency of traditional mathematics in modern scientific research (previous section), one can start by logically asking what may be essentially missing in that traditional mathematics approach and results, especially as compared to real processes and phenomena. It is not really difficult to see the probable answer: usual mathematics does not propose the consistent, unreduced solution to the arbitrary, real interaction problem, while it is the development of such many-component interaction processes that gives rise to all observed phenomena and structures (of increasingly complicated configuration for modern tasks).

As usual approach tries to look for smooth and single-valued, "analytical" or "exact" (in the narrow sense) solutions, it quickly discovers that such solutions for arbitrary interaction can be found only for the strongly limited situation of one-dimensional problem equivalent to interaction of only two material points. Any real interaction in the three-dimensional P a g e | 3 world, including large numbers of interacting components and their eigenmodes for real cases of interest, falls far outside of those limits of "integrable" problems within which that kind of explicit solution provides the adequate picture of reality.

All that remains then for this conventional approach in real many-body interaction cases is to use an approximation, or "model", where the real interaction problem is replaced with one or many integrable, effectively one-dimensional problems, including the "perturbation theory" description of deviations from real system behaviour (which cannot be correct and gives typical "divergent series" as a sign of unknown unreduced solution).

It is at this point that traditional mathematics makes its (expected) fatal reduction mistake by assuming too easily (without any reasonable substantiation) that the unreduced interaction problem solution and thus the entire picture of any process and structure dynamics would somehow represent at least a qualitatively similar version of those reduced, effectively one-dimensional model solutions, where further details can then be adjusted in the direction of observed features by mechanical parameter variation, perturbative modifications or simple heuristic guesses about the "real" solution. This reductive modelling approach, often justified by the subjectively appreciated "beauty" of the proposed abstract structures, leads to the observed growing inefficiency of traditional mathematics, and we shall rigorously specify the origin of these limitations in the next section.

(2) Unreduced solution. If one wants (and needs!) now to return to the natural beauty of the perceived, unreduced reality in its mathematically rigorous picture, then one must restart the entire mathematical enterprise, this time from the unreduced, non-simplified solution of real interaction problem, describing the emerging configuration and dynamics of all natural processes and structures, on any scale [START_REF] Kirilyuk | Theory of charged particle scattering in crystals by the generalized optical potential method[END_REF][START_REF] Kirilyuk | Quantum chaos and fundamental multivaluedness of dynamical functions[END_REF][START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Universal Symmetry of Complexity and Its Manifestations at Different Levels of World Dynamics[END_REF][START_REF] Kirilyuk | Dynamically Multivalued Self-Organisation and Probabilistic Structure Formation Processes[END_REF][START_REF] Kirilyuk | Complex-Dynamical Extension of the Fractal Paradigm and Its Applications in Life Sciences[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF][START_REF] Kirilyuk | The Last Scientific Revolution[END_REF]. It is not difficult to obtain a universal enough interaction problem formulation, conveniently provided by a generalised Hamiltonian equation for the system state-function, such as the Hamilton-Jacobi equation from classical mechanics or Schrödinger equation for quantum systems. We call it "existence equation" in the general case, as it does provide a simple description of the arbitrary system configuration, without any further assumptions. A more special case of open system (time-depending interaction) is easily included in the same starting formulation. Many other equations can eventually be reduced to a Hamiltonian formulation, and we self-consistently show in our further analysis of thus introduced interaction process that the results (forming the next-level system components) will always interact and behave according to the generalised, now well-specified Hamilton-Schrödinger formalism [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Universal Symmetry of Complexity and Its Manifestations at Different Levels of World Dynamics[END_REF][START_REF] Kirilyuk | Dynamically Multivalued Self-Organisation and Probabilistic Structure Formation Processes[END_REF][START_REF] Kirilyuk | Complex-Dynamical Extension of the Fractal Paradigm and Its Applications in Life Sciences[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF].

The key point of the problem is related to the fact that this unreduced equation for arbitrary real system with interaction (actually in any problem formulation) is "nonintegrable", i. e. it cannot be explicitly solved in that "closed" form of conventional "exact" solution.

One can approach the unreduced interaction problem solution by reformulating it in terms of so-called optical, or effective, potential method [START_REF] Kirilyuk | Theory of charged particle scattering in crystals by the generalized optical potential method[END_REF][START_REF] Kirilyuk | Quantum chaos and fundamental multivaluedness of dynamical functions[END_REF][START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF] originating in scattering theory and explicitly taking into account the complex, nonlinear interaction links development. One first uses the standard technique of problem expression in terms of eigen-solutions of its free components, transforming it to the equivalent (and equally nonintegrable) system of P a g e | 4

equations for respective state-function components. One tries then the substitution method for this system of equations using the Green's function technique and obtains a single equation for one state-function component of externally integrable form, but containing, instead of original interaction potential, the effective potential (EP) operator with the complicated nonlinear dependence on the problem solutions to be found, rendering the problem again nonintegrable, as should be expected for this equivalent transformation.

It is important, however, that this unreduced EP reveals the new, universal quality of the unreduced interaction problem solution designated as fundamental dynamic multivaluedness, or redundance, which can be properly, rigorously specified and after that helps to express the unreduced solution in a truly exact form accepting suitable, now properly correct approximations where necessary. This new quality of dynamic multivaluedness, constituting the key difference between the new and traditional mathematics frameworks, means that the problem has many equally real, locally complete and therefore mutually incompatible solutions called system realisations, each of them being generally similar to the unique solution for the emerging system configuration assumed in the effectively one-dimensional, dynamically single-valued approach of usual analysis. The redundant solution multiplication originates in the mentioned nonlinear dependence of the unreduced EP on the problem eigenvalues, which leads to the essential growth of the maximum eigenvalue power in the characteristic equation determining the total solution number [START_REF] Kirilyuk | Quantum chaos and fundamental multivaluedness of dynamical functions[END_REF][START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Universal Symmetry of Complexity and Its Manifestations at Different Levels of World Dynamics[END_REF][START_REF] Kirilyuk | Dynamically Multivalued Self-Organisation and Probabilistic Structure Formation Processes[END_REF][START_REF] Kirilyuk | Complex-Dynamical Extension of the Fractal Paradigm and Its Applications in Life Sciences[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF].

Being equally real but mutually incompatible, multiple system realisations thus obtained are forced, by the same driving interaction, to permanently replace each other, appearing and disappearing in causally random order thus rigorously (and dynamically) defined. We simultaneously obtain therefore the rigorously defined notions of event, (real) change, emergence and the well-specified origin of physically real, unstoppable (changing realisations) and irreversible (truly random realisation choice) time flow. The dynamically emerging, naturally discrete space elements are also rigorously obtained as respective eigenvalue separations for the unreduced EP formalism [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Universal Symmetry of Complexity and Its Manifestations at Different Levels of World Dynamics[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF].

Moreover, during each of system transitions between its plural "regular" realisations containing its dynamically entangled components (or degrees of freedom) the system is forced to transiently disentangle them within a special intermediate realisation, before their new, generally somewhat different dynamic entanglement within the next emerging, randomly chosen realisation. This intermediate, or "main", realisation with transiently disentangled, quasi-free components and chaotically fluctuating structure represents the universal extension of thus causally explained quantum-mechanical wavefunction and all statistical "distribution functions" at higher interaction levels (including also the wave-like brainfunction at the level of unconscious and conscious brain dynamics [START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF]). We therefore call the main system realisation thus explicitly obtained in the unreduced interaction analysis the generalised wavefunction, or distribution function, for any interaction level. It is naturally provided with the rigorously derived and physically transparent generalised Born rule, expressing the regular realisation emergence probability as the corresponding value of the generalised wavefunction (or its modulus squared for quantum and other wave-like levels).

While all these results are derived by the rigorous analysis of the unreduced interaction process in the generalised EP method [START_REF] Kirilyuk | Theory of charged particle scattering in crystals by the generalized optical potential method[END_REF][START_REF] Kirilyuk | Quantum chaos and fundamental multivaluedness of dynamical functions[END_REF][START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Universal Symmetry of Complexity and Its Manifestations at Different Levels of World Dynamics[END_REF][START_REF] Kirilyuk | Dynamically Multivalued Self-Organisation and Probabilistic Structure Formation Processes[END_REF][START_REF] Kirilyuk | Complex-Dynamical Extension of the Fractal Paradigm and Its Applications in Life Sciences[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF] (in two independent forms, algebraic and P a g e | 5 graphical [START_REF] Kirilyuk | Quantum chaos and fundamental multivaluedness of dynamical functions[END_REF][START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF]), one can provide also a simple transparent explanation of the origin and universality of dynamic multivaluedness by considering a schematic interaction picture between two, for example attracting, objects with N components (or dynamical modes) each. While in the ordinary, dynamically single-valued analysis one would obtain only the same number N of respective binary component or mode combinations, forming elemen- tary solutions (eigen-solutions), in the unreduced interaction analysis one takes into account the complete number 2 N of component or mode combinations, which gives the Nfold redundance with respect to the same number of "places" N for the interaction results. One obtains thus the unstoppable system realisation change in causally random order. This permanent realisation change implies also the omnipresent dynamic instability within any real system due to the unreduced and thus always self-amplifying interaction between multiple components and modes trying to "rearrange" the metastable configuration of any realisation. On the other hand, the fundamental dynamic discreteness, or quantisation, of realisations (and thus of any real system dynamics) is also finally due to this selfamplifying, holistic character of the unreduced interaction, which can stop its current tendency only around a metastable system configuration in a regular realisation (which provides another definition of the latter).

The derived causal, dynamic randomness of realisation emergence in any interaction process provides also the purely dynamic origin and a priori values of probabilities of realisation emergence [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Universal Symmetry of Complexity and Its Manifestations at Different Levels of World Dynamics[END_REF][START_REF] Kirilyuk | Dynamically Multivalued Self-Organisation and Probabilistic Structure Formation Processes[END_REF][START_REF] Kirilyuk | Complex-Dynamical Extension of the Fractal Paradigm and Its Applications in Life Sciences[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF]. As all elementary realisations are equally real, the probability r  of the r-th realisation emergence is given by

1 1 ,1 rr N r N          , (1a)
where N  is the total realisation number (determined by the number of suitable system component or their eigenmode combinations). Since for many real observations of "selforganised" dynamic regimes with densely packed groups of similar elementary realisations (see below) one deals rather with compound realisations containing many elementary ones, in the general case this dynamic probability definition takes the form 1,..., ; , 1

r r r r r rr N N N N N N                , (1b)
where r N is the number of elementary realisations within the r-th actually observed, com- pound realisation. This original definition of realisation probability correlates, of course, with the above generalised Born rule for practical calculation of the same dynamic probabilities from the generalised wavefunction ()

x  , 2 () rr x    .
It is important to emphasize that we obtain in this way the absolutely universal origin and meaning of randomness in this world, as well as the universal and dynamic meaning and values of related probability, extending essentially the respective formal, non-dynamical concepts of conventional science and mathematics. We see that randomness is omnipresent in real structures (though sometimes in a hidden , "self-organised" form, see below) and can only be of dynamic and genuine, "noncomputable" and "undecidable" character.

Correspondingly, the general, now truly complete solution of any real interaction problem, expressed in terms of measured system density, () x  , is obtained in the form of dynamical-

P a g e | 6
ly probabilistic sum of individual realisation densities { ()

r x  } [3-14]: 1 ( ) ( ) r N r xx       , (2)
where the sign  marks the special, dynamically probabilistic meaning of the sum implying that particular summands () r x



appear there probabilistically, with the corresponding dynamic probabilities r  , so that the result is a probabilistically fluctuating function. In the limit of long enough observations, with large numbers of realisation emergence events, the measured system density tends to the stationary expectation value:

    rr r xx      . ( 3
)
It is important, however, that contrary to conventional, dynamically single-valued mathematics, our description does not depend on any such "statistical" assumptions, so that the universal dynamic-probability expressions (1)-( 2) remain valid and well-defined for any single event of realisation emergence and even before any event happens at all.

Another qualitatively new feature of the unreduced problem solution (3) is the multilevel, probabilistically fractal structure of the complete realisation set, so that summations in ( 2) and ( 3) actually include a multilevel hierarchy of sums, especially for higher-complexity systems with many interacting components and eigenmodes. The dynamically multivalued fractal of the complete problem solution is explicitly obtained within the above unreduced EP method [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamical Extension of the Fractal Paradigm and Its Applications in Life Sciences[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF] and in general (except rare special cases) does not possess the simplified scale symmetry of usual, abstract fractals. It realises instead the absolutely exact and realistic universal symmetry of complexity (see below), including permanent probabilistic motion of fractal "branches", which provides the important property of dynamic adaptability and related huge power of unreduced interaction dynamics, underlying the "magic" features of life and intelligence [START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamical Extension of the Fractal Paradigm and Its Applications in Life Sciences[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF]. If we consider the most fundamental interaction level giving rise to the unified world structure emergence (attraction of two initially homogeneous protofields [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF]), then the emerging dynamically multivalued fractal of this process represents the single, dynamically unified and exact structure of the Universe.

We can now provide the universal definition of the main quantity of dynamic complexity C of any real structure or process in the form of any growing function of the number of system realisations or rate of their change, equal to zero for the (unrealistic) case of only one realisation (exclusively considered in the usual framework):

  , 0 , (1) 0 C C N dC dN C         . (4)
Examples include  

0 ln C N C N   ,   0 ( 1) C N C N  
, generalised energy-mass (tem- poral rate of realisation change) and momentum (spatial rate of realisation emergence) [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Universal Symmetry of Complexity and Its Manifestations at Different Levels of World Dynamics[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF]. It is evident that thus defined complexity automatically includes equally universally defined dynamic randomness, or chaoticity, due to the common underlying phenomenon of plural, equally real and mutually incompatible realisations.

We emphasize the universally positive value of this unreduced dynamic complexity of real, dynamically multivalued world structures of any level, starting already from (massive) elementary particles, in strong contrast to vague ideas of complexity in usual science, where it is "intuitively" attributed to externally "sophisticated" structures and phenomena. As to P a g e | 7

observed externally regular motions and rigid shapes of various objects, they represent but a particular dynamic regime of "self-organised" internal chaoticity of multiple, but similar realisations. In the general case of multi-level dynamics we deal here with the limit of dynamically multivalued SOC (self-organised criticality). In the opposite limit of sufficiently different realisations emerging with comparable probabilities we obtain the regime of global, or uniform, chaos. The rigorous analysis of the unreduced EP formalism provides the well-defined criterion of global chaos onset (the resonance between major system motions) and the gradual transition to the opposite limit of multivalued SOC, actually spanning the entire variety of observed dynamic regimes and behaviour patterns [START_REF] Kirilyuk | Quantum chaos and fundamental multivaluedness of dynamical functions[END_REF][START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Dynamically Multivalued Self-Organisation and Probabilistic Structure Formation Processes[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF].

Since the universal complexity thus defined is determined by the system initial configuration (through the total realisation number) it is conserved in any system structure emergence and transformation. Something, however, does change during interaction process development, and this change can be rigorously described as unstoppable transformation of the initial, potential (hidden) form of dynamic complexity, or dynamic information, I, to the final, unfolded-structure form of dynamic entropy, S, which preserves their sum, the total dynamic complexity C : C I S , Δ0 C  , Δ Δ 0 SI    . Thus obtained complexity conservation law includes the generalised and equally universal entropy growth law (extended second law of thermodynamics), which now describes also all cases of externally regular structure formation, thus solving various respective problems of conventional dynamically single-valued science [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Universal Symmetry of Complexity and Its Manifestations at Different Levels of World Dynamics[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF].

As this unified complexity conservation law describes the dynamic symmetry between system realisations practically implemented by system motion in real time, we call it the universal symmetry of complexity. It is not difficult to show that the extended, now universal quantity of action is a major integral measure of complexity, expressing the above dynamic information, and then one can obtain the differential expression of the universal symmetry of complexity in the form of extended Hamilton-Jacobi equation for action and the related generalised Schrödinger equation for the generalised wavefunction (intermediate realisation) [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Universal Symmetry of Complexity and Its Manifestations at Different Levels of World Dynamics[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF]. The obtained unified Hamilton-Schrödinger formalism is applicable to any system behaviour and represents the extended generalisation of various "model" dynamic equations. The underlying universal symmetry of complexity also unifies the extended versions of all known, usually postulated laws and principles [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Universal Symmetry of Complexity and Its Manifestations at Different Levels of World Dynamics[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF].

Note that the obtained intrinsically complete picture of the dynamically multivalued interaction process development is a qualitative and explicit extension of usual, dynamically single-valued description, including all conventional notions of "complexity", "chaoticity", "self-organisation", "attractors", "multistability", etc., which represent however intricate but always dynamically single-valued, point-like, zero-complexity projection of the unreduced, dynamically multivalued reality. We also call the entire content of that traditional dynamically single-valued projection unitary science (and paradigm), as it neglects the qualitatively inhomogeneous transitions between multiple system realisations and preserves only smooth pseudo-evolution (without real time) within only one realisation (often originating in the described intermediate realisation of the generalised wavefunction). In this way, the universal science of complexity presented here clearly specifies the origin and avoids various manifestations of the intrinsic incompleteness of usual mathematical framework of science, including the famous Gödel's incompleteness. P a g e | 8

(3) Extended mathematics of complexity. We can now provide a concise outline of proper mathematical novelties of the universal science of complexity thus obtained as a result of unreduced solution of arbitrary many-body interaction problem [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex-Dynamical Extension of the Fractal Paradigm and Its Applications in Life Sciences[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | The Last Scientific Revolution[END_REF]: (i) Non-uniqueness of any real problem solution, in the form of fundamental dynamic multivaluedness (redundance) of system realisations, as opposed to conventional uniqueness theorems and solution type. Note the essential difference of our dynamic, interaction-driven multivaluedness from usual multivalued functions, as well as various unitary imitations of "multistability" within the single solution.

(ii) Omnipresent, universal, dynamic and genuine randomness due to equally real and incompatible realisation change in causally random order, providing clear understanding of usual vague notions of nonintegrability, nonseparability, noncomputability, uncertainty (indeterminacy), undecidability, stochasticity, broken symmetry, free will, etc. Truly regular structures, motions and patterns are absent, while they exclusively prevail in traditional mathematical framework, including its dynamically single-valued imitations of randomness and chaoticity. (iii) The absence of self-identity,  AA , for any structure A , tacitly assumed in traditional mathematics. In real world and in the new mathematics of complexity we have instead  AA (cf. eq. ( 2)), giving permanent irreversible change and causal time flow. (iv) Fractally structured multivalued dynamic entanglement of interacting system components in the unreduced problem solution, providing the rigorous mathematical definition of the perceived quality (or texture) of emerging structures, as opposed to purely abstract, "immaterial" character of usual mathematical structures and models.

(v) Dynamic discreteness, or causal quantisation, of the unreduced interaction results and dynamics (and thus any real structure and process) eventually due to its holistic character, giving rise to qualitatively inhomogeneous, nonunitary system evolution, with the opposite quality of the traditional mathematical framework.

One should add to these features the essentially, dynamically unified character of the new mathematics of complexity expressed by its single, unified structure of dynamically probabilistic fractal and single, unified law of the universal symmetry of complexity, which give rise to the entire, now dynamically unified variety of all particular structures, objects, laws and principles. In particular, one obtains the naturally unified picture of reality at the fundamental, lowest complexity levels of elementary particles and fields, where this unification includes the causally specified dynamic origin of particles, their properties, fundamental interaction forces and constants, quantum and relativistic behaviour [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF].

Applications and conclusion: The unlimited efficiency of the extended mathematics of complexity

We can only briefly refer to various already realised applications of the universal science of complexity at different complexity levels, from the unified, causally complete fundamental physics to life sciences, intelligence and consciousness, complex computer systems, sustainability and civilisation development [START_REF] Kirilyuk | Theory of charged particle scattering in crystals by the generalized optical potential method[END_REF][START_REF] Kirilyuk | Quantum chaos and fundamental multivaluedness of dynamical functions[END_REF][START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Universal Symmetry of Complexity and Its Manifestations at Different Levels of World Dynamics[END_REF][START_REF] Kirilyuk | Dynamically Multivalued Self-Organisation and Probabilistic Structure Formation Processes[END_REF][START_REF] Kirilyuk | Complex-Dynamical Extension of the Fractal Paradigm and Its Applications in Life Sciences[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF][START_REF] Kirilyuk | The Last Scientific Revolution[END_REF], confirming its problem-solving power, which clearly originates just in the described "dynamically multivalued" extension of traditional dynamically single-valued framework.

P a g e | 9

The critically acute modern problems of "complexity crisis" in both fundamental science and real world development (see the first section) find thus their universally applicable solution within the framework of extended mathematics of complexity presented above, and we also show why exactly they cannot be solved within the artificially restricted unitary science paradigm. All the real-world structures are obtained as a result of unreduced interaction processes, and it is just the right moment now to extend our scientifically rigorous description of the emerging strong-interaction, "globalised" reality to the causally complete, provably reliable understanding of the underlying complex, multivalued and intrinsically creative interaction dynamics.

P a g e | 12

proach can provide either "surprisingly" realistic or strangely "mysterious" image of the multivalued real system feature depending on the more or less successfully guessed "projection kind/direction" (becoming much less obvious for more complex systems).

The transition from the traditional, artificially simplified description to the proposed intrinsically complete mathematical framework corresponds thus to the transition from inevitably separated and often "strangely looking" usual projections to the full-dimensional and therefore causally complete, dynamically unified image of unreduced reality (see the essay for details and references). That (mathematically specified) vision shows that the "extreme" program of the ultimately complete and unified science can be quite realistic and natural (as opposed to the accumulating pessimism of the traditional science framework).

Appendix 2: From Postulated Being to Dynamical Becoming

The growing loss of certainty of modern, apparently successful mathematics is largely due to its tacitly increasing departure from the fundamental principle of internal consistency closely related to the demand of realism understood as a close enough correspondence to the physically consistent natural structures this mathematical framework of science is supposed to efficiently describe.

In the same way as the elements of physical reality interact and give rise to ever more elaborated structures and phenomena, the "abstract" elements of mathematical framework progressively form respective abstract structures and laws, according to rigorous derivation rules ensuring the result consistency.

However, another, qualitatively more abstract approach to mathematical description of reality has appeared and quickly grown especially in the age of "new physics" starting from the beginning of the 20th century, where the "final", resulting mathematical structures and laws are somehow heuristically "guessed" and then simply postulated for further direct "confirmation by experiment", but without the real, consistent derivation and logical understanding of their origin. In particular, their growing number appears even to be quite inconsistent, i. e. incomplete, contradictory and "mysterious" (quantum mechanics, relativity, gravity, "dark matter", "hidden dimensions", "broken symmetries", etc.), but still "unreasonably efficient" and "confirmed by experiment". This tendency of "mathematical physics" is reduced to never-ending trial-and-error attempts of direct postulation of a new mathematical "language" or "manifold" with its following "adaptation" to a limited set of "experimental observations". No derivation and solution of a dynamic equation from first principles is involved any more in this now dominating approach (contrary to previous situation of few postulates and the majority of derived features), which strangely and increasingly resembles the ancient Ptolemaic kind of "exact science".

Such kind of mathematical description is the ultimate expression of the "modelling" logic from the previous Discussion Abstract, while the principle of internal consistency can also be formulated as self-referential, autonomous and "automatically correct" nature of the desired truly consistent and complete mathematical framework. The self-referential, "living" and "intelligent" framework of reality-based mathematics should naturally include the intrinsically creative, "becoming" character of physical reality and its "biting" time-change P a g e | 13

(as emphasized especially by Prigogine and his school and previously by Bergson), as opposed to only "being", mechanistically fixed and effectively timeless character of mainstream postulated structures.

Needless to say, the necessary qualitative transition from such postulated "Being" of dead abstract models to dynamically creative, intrinsically complete "Becoming" of consistently derived structures is accomplished with the help of unreduced interaction problem solution presented in this essay, due to its clearly specified extended mathematical features of dynamic multivaluedness, entanglement and probabilistic fractality (summarised by the universal dynamic complexity). Wouldn't it represent the best possible kind of natural (and otherwise increasingly missing) unification of the mathematical and physical realities and universes, without any limits on any high-level applications?

Appendix 3: Conceptual Mathematics and Causally Complete Science

If mathematics is not only a technical tool or language of knowledge, but is designed for a deeper role of consistent science basis, then it follows that irrespective of details it should contain special foundational directions, or conceptual mathematics, which should use the exact description methods specifically for essential extension and completion of scientific knowledge, rather than only technical description of new structures and phenomena at the same level of understanding.

While particular examples of such kind of fundamentally new features can eventually be found (e. g. non-Euclidean geometry or chaotic dynamics), this quality is rarely emphasized as the necessary intrinsic feature of mathematical framework, apart from applications. Moreover, the appearing foundational novelties of modern mathematics show quickly growing trial-and-error uncertainty (cf. Kline, 1980) compromising their real efficiency as a basis for essential science progress, in contrast to spectacular advances of technical tools and methods as such. Given the above starting assumption of a deeper role of mathematics and the ensuing universality of conceptual mathematics, one arrives at the necessity of essential upgrade of the latter confirmed by the resulting qualitative completion of scientific knowledge.

The extended mathematics of complexity presented in this essay provides a well-specified realisation of this absolutely general conclusion, thus not only solving particular problems, but also introducing the necessary, otherwise increasingly missing creative order into the entire huge system of empirical, technical-theoretic and conceptual-theoretic knowledge.

As a result of this proper extension of conceptual mathematics, one obtains a whole series of knowledge completion and unification features. Not only fundamental physics regains its otherwise increasingly broken unity, causality and consistency, but now it can also be correctly unified with the equally rigorous description of all higher levels of reality to form a single, universal and intrinsically comprehensible form of absolutely exact scientific knowledge (see the essay and its references for details). We also destroy thus the familiar annoying barrier between the technically rigorous description and transparent figurative understanding of reality merging now into the higher-level, causally complete and widely understandable kind of science.

P a g e | 14

In order to see the advantages of thus conceptually extended mathematics, one can compare this really emerging unification with the growing serious doubts of conventional science professionals about not only any possibility of unified knowledge (e. g. Gleiser 2010, 2010a, 2011, 2013, Wells, 2013, Noë, 2014), but even the very existence of truly universal and rigorous scientific laws (Kauffman, 2011(Kauffman, , 2013)).

The resulting ultimate scientific revolution puts therefore an end to usual, "antagonistic" scientific revolutions (Kuhn, 1980) and leads to permanently progressing kind of intrinsically complete knowledge, where the notorious internal, qualitative "incompleteness" of traditional science and mathematics is replaced by only external, "quantitative" limitations of currently accessible observation space (or the "infinity problem").

Appendix 4: From Complexity Crisis to the New Science Age

The accumulating "difficult", practically "unsolvable" and therefore eventually destructive problems of today's global civilisation, from economical crisis to conflicting interests and missing sustainability, have something deep in common: the "untreatable" dynamic complexity of unreduced multicomponent interaction processes or systems. The same kind of impasse occurs in (traditional) fundamental physics, from the persisting old and growing new "mysteries" (quantum mechanics, unification, dark matter/energy, time, …) to the vital issues of sustainable energy sources and intelligent machines of all levels.

As shown in the present essay, usual mathematical framework of science can not be efficient in resolution of this omnipresent and dangerously growing complexity crisis, already because it does not propose the unreduced many-body problem solution, replacing it with various qualitatively limited point-like projections, or "models". We then reveal the origin and dynamical structure of the causally complete, dynamically multivalued problem solution and specify the new mathematical and physical qualities it discovers in real system origin and dynamics. It is due to those rigorously expressed and universally applicable new qualities that the obtained new mathematics of unreduced dynamic complexity can form the reliable basis for the complete solution of complexity crisis problems and further, now unlimited progress at the emerging superior levels of civilisation complexity.

It is important that the same new mathematics of unreduced complexity provides the causally complete and intrinsically unified description of fundamental physical systems, from elementary particles to the universe structure and evolution, now without postulated "mysteries", "dark" matters and other "hidden dimensions". Equally successful applications at all higher complexity levels confirm the proposed extended framework universality and put an end to the growing deep doubts of mainstream science about the power of scientific method as such (see also the end of the previous Discussion Abstract here).

In summary, the proposed new mathematics of unreduced complexity, unified by the universal symmetry of complexity of the dynamically probabilistic fractal, paves the way from the current deep complexity crisis to the new, basically unlimited civilisation progress driven by the extended, causally complete knowledge of the universal science of complexity.

P a g e | 10

Appendix 1: * To Be or Not to Be

In relation to the current situation in fundamental science, the estimates of the role and state of rigorous, mathematical description of reality apparently evolve towards two diverging groups.

One of them, the "mainstream mathematical physics", defends the existing, traditional development of mathematical description of reality by always simplified but internally technically powerful modelling, with its "unreasonable" successes and not less mysterious failures, "unsolvable" problems and accumulating "dark matters". One considers in this mainstream paradigm that one can continue to solve a large enough part of fundamental and practical problems with if not "unreasonable", but at least sufficiently high efficiency and precision, while the remaining "mysteries" (e. g. of time, quantum mechanics, or dark matter), "unsolvable" problems (real interaction) and "non-computable" phenomena (e. g. from the humanities) can be accepted in the form of "inexplicable postulates" or basically empirical and only mechanistically quantifiable knowledge.

In the second attitude, one considers that the truly consistent, not only rigorous, but also causally complete and unified description of reality is still possible, certainly beyond traditional limited "models" but within a qualitatively extended mathematical framework. Actually this is a "strong version of science" as objectively reliable form of knowledge, where one accepts the challenge of ultimately complete science covering eventually all knowledge and providing the totally consistent picture of reality.

The present essay describes a working version of the second approach, in the form of Universal Science of Complexity confirmed by various applications, from elementary particles to all high-level systems (biology, society, consciousness), now within the causally complete and totally unified description, with dynamically emerging, physically real space, time, intrinsic properties and laws. It is obtained as explicit extension of usual mathematics framework, in the form of dynamically multivalued unreduced solution to arbitrary real interaction problem, while the traditional theory modelling corresponds to the dynamically single-valued, effectively zero-dimensional (point-like) projection of the unreduced dynamics of any real system or process.

That explicit projection relation between the traditional and new mathematics provides also a transparent explanation of the "unreasonable efficiency" of the former strangely intermingled with its unsolvable problems, persisting mysteries and simply rigorously indescribable phenomena. Indeed, any example of geometrical projection of a threedimensional object to lower-dimension spaces shows immediately that the limited image of low-dimensional projection can vary essentially in its correspondence to the unlimited three-dimensional prototype, depending on the direction of projection "view" with respect to essential structural features of the object (consider a pencil projection varying from a realistic "rod" to the ambiguous "thick point"). In the same way, the ultimately limited point-like dynamical projection of real system behaviour within the traditional model ap-


Philippe Jaming
email: philippe.jaming@univ-orleans.fr

NAZAROV'S UNCERTAINTY PRINCIPLES IN HIGHER DIMENSION

Keywords: 1991 Mathematics Subject Classification. 42B10 strong annihilating pair, random periodization, uncertainty principle

In this paper we prove that there exists a constant C such that, if S, Σ are subsets of R d of finite measure, then for every function

where f is the Fourier transform of f and w(Σ) is the mean width of Σ. This extends to dimension d ≥ 1 a result of Nazarov [Na] in dimension d = 1.

Introduction

An uncertainty principle is a mathematical result that gives limitations on the simultaneous localization of a function and its Fourier transform. There are many statements of that nature, the most famous being due to Heisenberg-Pauli-Weil when localization is measured in terms of smallness of dispersions and to Hardy when localization is measured in terms of fast decrease of the functions. We refer the reader to the surveys [FS, BD] and to the book [HJ] for further references and results.

We will need a few notations before going on. In this paper d will be a positive integer, all subsets of R d considered will be measurable and we will denote by |S| the Lebesgue measure of S. The Fourier transform is defined for

f ∈ L 1 (R d ) ∩ L 2 (R d ) by f (ξ) = R d f (x)e 2iπ x,ξ dx
and extended to all of L 2 (R d ) in the usual way.

In this paper, we are interested in another criterium of localization, namely smallness of support. For instance, it is well known that if a function is compactly supported, then its Fourier transform is an entire function and can therefore not be compactly supported. We may then ask what happens if a function f and its Fourier transform f are only small outside a compact set? This leads naturally to the following definition:

Definition.

Let S, Σ be two Borel subsets of R d . Then we will say that -(S, Σ) is an annihilating pair (a-pair in short) if the only function f that is supported in S and such that its Fourier transform f is supported in

Σ is f = 0; -(S, Σ) is a strong annihilating pair (strong a-pair in short) if there exists a constant C = C(S, Σ) such that for every f ∈ L 2 (R d ), R d |f (x)| 2 dx ≤ C R d \S |f (x)| 2 dx + R d \Σ | f (x)| 2 dx .
This notion has been extensively studied in the case S is a compact set by Logvinenko and Sereda [LS], Paneah [START_REF] Paneah | Support-dependent weighted norm estimates for Fourier transforms[END_REF][START_REF] Paneah | Support-dependent weighted norm estimates for Fourier transforms[END_REF], Havin and Jöricke [HJ] and Kovrijkine [Ko], see also [HJ]. In this case the class of all Σ's for which (S, Σ) is a strong a-pair is characterized. Moreover, if S is convex, there are fairly good estimates of the constant C(S, Σ) in terms of the geometry of S and Σ.

For sets S, Σ that are sublevel sets of quadratic forms, the problem has been studied by Shubin, Vakilian, Wolff [SVW] and by Demange [START_REF] Demange | Principes d'incertitude associés à des formes quadratiques non dégénérées[END_REF][START_REF] Demange | Inégalités d'incertitude associées à des fonctions homogènes[END_REF].

Here we will focus on the case of S, Σ being of finite Lebesgue measure. This was first studied by Benedicks [Be] who proved that in this case (S, Σ) is an a-pair, and a little abstract nonsense allows to prove that in this case (S, Σ) is also a strong a-pair, see [BD]. This last fact was proved with a different method by Amrein and Berthier [AB]. Unfortunatly both proofs do not give any estimate on the constant C(S, Σ). By using a randomization of Benedicks proof and an extension of a lemma of Turan, Nazarov [Na] showed that in dimension 1, the constant is of the form C(S, Σ) = Ce C|S||Σ| . It was thought for some time that Nazarov's method would extend to higher dimension to give a constant of the same form. This is far from the expected optimal which is thought to be obtained by taking S, Σ balls of radius R and f a Gaussian function, which gives

C(S, Σ) = Ce CR 2 = e C(|S||Σ|) 1/d .
The aim of this paper is to push Nazarov's technique as far as possible and thus improve the Ce C|S||Σ| constant when the geometry of Σ is suitable. Using the recent extension of Nazarov's Turan lemma to higher dimension by Fontes-Merz [FM], we will prove the following result:

Theorem. There exists a constant C such that, for every sets S, Σ ⊂ R d of finite Lebesgue measure and for every

f ∈ L 2 (R d ), R d |f (x)| 2 dx ≤ Ce C min |S||Σ|,|S| 1/d w(Σ),w(S)|Σ| 1/d R d \S |f (x)| 2 dx + R d \Σ | f (x)| 2 dx
where w(Σ) is the mean width of Σ.

In particular, if S or Σ has a geometry that is close to a ball, this is in accordance with what is supposed to be the optimal result. The remaining of this paper is devoted to the proof of this theorem. In order to do so, we first extend to higher dimension the random periodization technique. Then we recall the Turan type estimates we will need. The last section is then devoted to the proof of the theorem.

Random Periodization

f ∈ L 1 (R d ) SO(d) +∞ 0 f v ρ(u) v d-1 dv dν d (ρ) = C R d f (x) dx.

The higher dimensional Lattice Averaging Lemma.

The following lemma was proved by Nazarov in dimension d = 1.

Lemma 2.1 (Lattice Averaging Lemma). Let d ≥ 1 be an integer, then for every ϕ ∈ L 1 (R d ), ϕ ≥ 0, the following estimates hold

SO(d) 2 1 k∈Z d \{0} ϕ v ρ(k) dv dν d (ρ) ≃ x ≥1 ϕ(x) dx and (2.1) SO(d) 2 1 k∈Z d \{0} ϕ ρ(k) v dv dν d (ρ) ≃ x ≥1/2 ϕ(x) dx.
Here, as usual, by A ≃ B we mean that there exists a constant C depending only on d such that

1 C B ≤ A ≤ CB.
Proof. With (2.1), we get

SO(d) 2 1 k∈Z d \{0} ϕ v ρ(k) dv dν d (ρ) ≃ k∈Z d \{0} SO(d) 2 1 ϕ v k ρ(k/ k ) v d-1 dv dν d (ρ) = C k∈Z d \{0} 1≤ x ≤2 ϕ( k x) dx = C k∈Z d \{0} 1 k d k ≤ x ≤2 k ϕ(x) dx = C x ≥1 ϕ(x) k ≤ x ≤2 k 1 k d dx ≃ x ≥1 ϕ(x) dx since, for x ≥ 1, k ≤ x ≤2 k 1 k d ≃ u ∈ R d : x /2 ≤ u ≤ x x d = |B(0, 1) \ B(0, 1/2)|.
For the second statement, one first changes v into 1/v and the remaining of the proof is similar.

Definition.

For a function f ∈ L 2 (R), ρ ∈ SO(d) and v > 0, we define the periodization

Γ ρ,v (t) = Γ ρ,v (f )(t) of the function f by Γ ρ,v (t) = 1 √ v k∈Z d f ρ(k + t) v .
The series in the definition of Γ ρ,v converges in L 2 (T d ) and represents a periodic function. An easy computation shows that the Fourier coefficients of Γ

ρ,v are Γ ρ,v (m) = √ v f v t ρ(m) for m ∈ Z d . Notation.
In the sequel, v will be considered as a random variable equidistributed on the interval (1, 2) and ρ as a random variable equidistributed on SO(d). The expectation with respect to these random variables will be denoted by E ρ,v 2.3. Properties of random periodizations. From the Lattice Averaging Lemma we shall derive the following simple but useful properties of the random periodization.

Proposition 2.2. Let d ≥ 1 be an integer and C = C(d) be the constant defined in Lemma 2.1. Let S ⊂ R d be a set of finite measure and let f ∈ L 2 (R d ) be supported in S.

Then (i) for all v ∈ (1, 2), |{ t ∈ (0, 1) : Γ ρ,v (t) = 0 } ≤ 2 d |S|; (ii) E ρ,v Γ ρ,v 2 L 2 (0,1) ≤ 2| f (0)| 2 + 2C f 2 L 2 (R d ) ≤ 2(|S| + C) f 2 L 2 (R d ) .
Proof. i) The set of all points t ∈ [0, 1] d for which the summand f ρ(k+t) v in the series defining Γ ρ,v

does not vanish equals

v t ρ(S) ∩ [0, 1] d + k . Therefore, |{t ∈ [0, 1] d : Γ ρ,v (t) = 0}| ≤ k∈Z d |v t ρ(S) ∩ [0, 1] d + k | = |v t ρ(S)| ≤ 2 d |S|.
ii) Parseval's Identity gives

E ρ,v Γ ρ,v 2 L 2 (T d ) = E ρ,v   k∈Z d | Γ ρ,v (k)| 2   = E ρ,v | Γ ρ,v (0)| 2 + E ρ,v   k∈Z d \{0} | Γ ρ,v (k)| 2   . But | Γ ρ,v (0)| 2 = v| f (0)| 2 ≤ 2| f (0)| 2
, and, with the Lattice Averaging Lemma,

E ρ,v   m∈Z d \{0} |Γ ρ,v (m)| 2   = SO(d) 2 1   m∈Z d \{0} v| f v ρ(m) | 2   dv dν d (ρ) ≤ 2 SO(d) 2 1   m∈Z d \{0} | f v ρζ(m) | 2   dv dν d (ρ) ≤ 2C R d | f ρ(ξ) | 2 dξ = 2C f 2 L 2 (R d ) .
It remains to notice that

| f (0)| 2 = S f (x)dx 2 ≤ |S| S |f (x)| 2 dx = |S| f 2 L 2 (R) .
Definition.

Let Σ ⊂ R be a measurable set with, 0 ∈ Σ. We consider the lattice

Λ = Λ(ρ, v) := {v t ρ(j) : j ∈ Z d } and denote M ρ,v = {k ∈ Z d : v t ρ(k) ∈ Σ} = Λ ∩ Σ.
Proposition 2.3. With the previous notations

(i) E ρ,v card M ρ,v -1 ≤ C|Σ|, in particular M ρ,v is almost surely finite; (ii) E ρ,v   m∈Z d \Mρ,v | Γ ρ,v (m)| 2   ≤ 2C R d \Σ | f (ξ)| 2 dξ. Proof. i) Since card M ρ,v = 1 + m∈Z d \{0} χ Σ v t ρ(m) , we have E ρ,v card M ρ,v -1 = SO(d) 2 1 k∈Z d \{0} χ Σ v t ρ(k) dv dν d (ρ) ≤ C R d χ Σ (x) dx = C|Σ|. ii) From the expression of Γ ρ,v we get that E ρ,v   m∈Z d \Mρ,v | Γ ρ,v (k)| 2   is = SO(d) 2 1   m∈Z d \{0} v f v t ρ(m) 2 χ R d \Σ v t ρ(m)   dv dν d (ρ) ≤ 2 SO(d) 2 1   m∈Z d \{0} f v t ρ(mk) 2 χ R d \Σ v t ρ(m)   dv dν d (ρ) ≤ 2C R d | f (ξ)| 2 χ R\Σ (ξ) dξ = 2C R d \Σ | f (ξ)| 2 dξ.
by Lemma 2.1.

A Turan Lemma

3.1. Nazarov and Fontes-Merz' Turan Lemmas.

For sake of completeness, we will recall here the Turan type estimates of trigonometric polynomials we will need.

Theorem (Nazarov's Turan Lemma [Na]) The original theorem of Turan deals with sets E that are arcs. The extension to higher dimension has been obtained in [FM] using a clever induction on the dimension.

Let P (t) = m k=1 c k e 2iπr k t with c k ∈ C \ {0}, r 1 < • • • < r m ∈ Z,
Corollary (Fontes-Merz's Turan Lemma [FM]) Let d ≥ 1 be an integer and let

p(z 1 , . . . , z d ) = m1 k1=0 • • • m d k d =0 c k1,...,k d z r 1,k 1 1 • • • z r r d ,k d d with r i,ki ∈ Z be a polynomial in d variables. Then, for every measurable set E ⊂ T d , sup z∈T d |p(z)| ≤ 14d |E| m1+•••+m d sup z∈E |p(z)|.
The quantity m 1 + • • • + m d is called the order of p (with the usual convention that we take the most compact possible representation of p) and is denoted by ord p. In general

m 1 + • • • + m d ≤ d max m i ≤ d Card Spec p while Card Spec p ≤ (m 1 + 1) • • • (m d + 1).

An estimate of the average order.

The notion of order of a polynomial suggests the following definition of the order of a subset of Z d . Definition. Let M ⊂ Z d be a finite set, we will say that M is of order k and write ord M = k if there exists integers m 1 , . . . , m d with m 1 + • • • + m d = k such that the projection of M on the i-th coordinate axis has m i elements.

Finally, if Λ = AZ d is a lattice and M ⊂ Λ is finite, we will call ord M = ord A -1 M .

Note that

m 1 = k∈Z sup k ′ ∈Z d-1 χ M (k, k ′ )
with similar expressions for the other m i 's.

In order to estimate the order of the set M ρ,v introduced before Proposition 2.3, the easiest is to bound the order by the cardinal of the set, which amounts to bounding the supremum by the sum over k ′ ∈ Z d-1 in the above expression. One then gets E ρ,v ord M ρ,v -d ≤ C|Σ|. This shows in particular that it is enough to estimate this quantity when Σ is a relatively compact open set.

The proof of the uncertainty principle in the next section will then give a constant Ce C|S||Σ| in Nazarov's result. We will slightly improve this. in order to do so, let us introduce the following quantities:

-the average width: for a relatively compact open set Σ and for ρ ∈ SO(d), let P ρ (Σ) be the projection of Σ on the span of ρ(1, 0, . . . , 0). We define

w(Σ) = SO(d) |P ρ (Σ)| dν d (ρ)
the average width of Σ. If Σ is a ball, this is just its diameter.

-let us also introduce the measure µ on R d defined by

µ(Σ) = inf i∈I min(r i , r d i ) : {B(x i , r i )} i∈I is a cover of Σ .
Note that µ(Σ) ≤ C|Σ| since the d-dimensional Hausdorff measure is the Lebesgue measure.

We will now prove the following:

Proposition 3.1.
Let Σ be a relatively compact open set with 0 ∈ Σ. We consider a random lattice

Λ = Λ(ρ, v) := {v t ρ(j) : j ∈ Z d } and denote M ρ,v = {k ∈ Z d : v t ρ(k) ∈ Σ} = Λ ∩ Σ. Then E ρ,v ord M ρ,v -d ≤ C min µ(Σ), w(Σ) . Proof. Let m ρ,v (Σ) = k∈Z\{0} sup k ′ ∈Z d-1 χ Σ v t ρ(k, k ′ ) .
It is enough to prove that

(3.3) E ρ,v m ρ,v (Σ) ≤ C min µ(Σ), w(Σ) . As pointed out above, E ρ,v m ρ,v (Σ) ≤ C|Σ|. In particular, if Σ is a ball of radius r, E ρ,v m ρ,v (Σ) ≤ Cr d On the other hand m ρ,v := m ρ,v Σ ≤ k∈Z\{0} sup y∈R d-1 χ Σ t ρ(vk, y)
and the one-dimensional lattice averaging lemma then gives

E ρ,v (m ρ,v ) ≤ SO(d) |x|≥1 sup y∈R d-1 χ B ρ(a),r (x, y) dx dν d (ρ) ≤ C SO(d) |x|≥1 χ Pt ρ (x) dx dν d (ρ) ≤ Cw(Σ).
In particular, if Σ is a ball of radius r, then

E ρ,v m ρ,v (Σ) ≤ Cr. To conclude, it is enough to note that m ρ,v (Σ ∪ Σ ′ ) ≤ m ρ,v (Σ) + m ρ,v (Σ ′ ) and that if Σ ⊂ Σ ′ then m ρ,v (Σ) ≤ m ρ,v (Σ ′ ).
Covering Σ with balls then gives the desired result.

The result above is essentially sharp as the following example shows. For simplicity, we will give the example in dimension d = 2. Let N ≥ 1 be an integer and let R ≫ 1 be two real numbers. Let

Σ N = N -1 j=0 B R(cos 2π N j, sin 2π N j), 1 2 .
That is, Σ N is the union of N discs regularily placed on a big circle, see the figure below.

Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ '! &" %# $ '! &" %# $ '! &" %# $ '! &" %# $ '! &" %# $ '! &" %# $ '! &" %# $ '! &" %# $ '! &" %# $
The set Σ N : Note that each line orthogonal to a line through the origin meets at most two circles. Moreover, these circles have radius ≤ 1/2 thus, for k fixed, at most two segments { t ρ(vk, vk ′ ), v ∈ (1, 2)} can intersect Σ N . Therefore, the sup k ′ in the formula defining m 1 can be bounded below by

1 2 k ′ . Then, for each k, #{k ′ ∈ Z : t ρ(vk, vk ′ ) ∩ Σ N = 0} ≤ 2 thus m ρ,v Σ N ≥ 1 2 k∈Z\{0} k ′ ∈Z χ ΣN t ρ(vk, vk ′ ) ≃ |Σ N | ≃ N ≃ w(Σ N )
with the Latice Averaging Lemma.

Conclusion

The remaining of the proof follows the path of Nazarov's original argument. We include it here for sake of completeness.

Let us write ν(Σ) = min w(Σ), µ(Σ) . First, it is enough to prove that there exists a constant

C = C(d) such that Σ | f (ξ)| 2 dξ ≤ Ce Cν(|S| 1/d Σ) R d \Σ | f (ξ)| 2 dξ
for every f ∈ L 2 (S). Moreover, using a scaling argument, it is enough to show that, if |S| = 2 -d+1 , then for every set Σ and every f ∈ L 2 (S),

Σ | f (ξ)| 2 dξ ≤ Ce Cµ(Σ) R d \Σ | f (ξ)| 2 dξ. Set Γ ρ,v (t) = Γ ρ,v (f )(t) the random periodization of f . Then, setting E ρ,v = {t ∈ (0, 1) : Γ ρ,v (t) = 0}, we have by Proposition 2.2 i) that |E ρ,v | ≥ 1 -2 d |S| = 1 2 . Next, set M ρ,v := {m ∈ Z d : v t ρ(m) ∈ Σ ∪ {0}} and decompose Γ ρ,v = P ρ,v + R ρ,v where P ρ,v (t) = m∈Mρ,v Γ ρ,v (m)e 2iπmt while R ρ,v (t) = m∈Z d \Mρ,v Γ ρ,v (m)e 2iπmt .
By Proposition 2.3 ii),

E ρ,v R ρ,v 2 L 2 (0,1) = E ρ,v   m∈Z d \Mρ,v | Γ ρ,v (m)| 2   ≤ 2C R d \Σ | f (ξ)| 2 dξ, hence E ρ,v R ρ,v 2 L 2 (0,1) > 4C R d \Σ | f (ξ)| 2 dξ < 1 2 .
On the other hand, by Proposition 3.1,

E ρ,v ord P ρ,v ≤ Cµ(Σ) + d and therefore E ρ,v ord P ρ,v > 2 Cµ(Σ) + d) < 1 2 .
We thus get that the two events

(1) R ρ,v 2 L 2 (0,1) ≤ 4C R d \Σ | f (ξ)| 2 dξ,
(2) ord P ρ,v ≤ 2 Cµ(Σ) + d) happen simultaneously with non-zero probability, while the two events

(3) |E ρ,v | ≥ 1 2 , (4) | f (0)| 2 ≤ | P ρ,v (
0)| 2 are certain. We will now take v ∈ (1, 2), ρ ∈ S d-1 such that all four events hold simultaneously.

Further, by definition Γ

ρ,v = 0 on E ρ,v , that is P ρ,v and -R ρ,v coincide on E ρ,v . It follows that Eρ,v |P ρ,v (x)| 2 dx = Eρ,v |R ρ,v (x)| 2 dx. Hence x ∈ E ρ,v : |P ρ,v (x)| 2 ≥ 16C R d \Σ | f (ξ)| 2 dξ ≤ 1 4 and, as |E ρ,v | ≥ 1 2 , we get that | Ẽρ,v | ≥ 1 4 where Ẽρ,v =    x ∈ E ρ,v : |P ρ,v (x)| ≤ 4 C R d \Σ | f (ξ)| 2 dξ 1 2    .
We can now apply Turan's Lemma and get The values of the constants may be tracked and linked to those of the Random Averaging Lemma, but we do not expect these constants to be any near to optimal (as they are already not optimal in dimension 1) so we will not pursue this.

| f (0)| 2 ≤ | P ρ,v (
Note also that, with mutadis mutandis the same proof as in [Na] we obtain the following corollary:

Corollary 4.1.

Let S, Σ be two measurable subsets of R d and let C be the constant of the main theorem. Then, for every p ∈ (0, 2) and every f ∈ L p (R d ) with spectrum in Σ,

f p L p ≤ Ce Cp|S||Σ| R d \S |f (x)| p dx.

Aknowledgements

The author wishes to thank B. Demange, L. Grafakos for valuable conversations concerning the content of this paper and A. Bonami for a careful reading of the manuscript.

This work was supported by the Hungarian-French Scientific and Technological Governmental Cooperation, no. F-10/04

  2.1. Preliminaries. For any integer d, let SO(d) denote the group of rotations on R d . Denote by dν d the normalized Haar measure on SO(d). Then there exists a constant C = C(d) such that, for every u ∈ S d-1 , the unit sphere S d-1 of R d , and every function

  be a trigonometric polynomial of order ord P = m and let E be a measurable subset of T. Then

≤

  Ce Cν(Σ) R d \Σ | f (ξ)| 2 dξ.If we now apply this to f y (x) = f (x)e -2iπxy instead of f and to the set Σ y = Σ -y instead of Σ, we obtain that| f (y)| 2 ≤ Ce Cν(Σ) R d \Σ | f (ξ)| 2 dξ.and integrating this over Σ givesΣ | f (y)| 2 dy ≤ C|Σ|e Cµ(Σ) R d \Σ | f (ξ)| 2 dξ as claimed. 2