This homepage addresses the three Millennium problems A. The Riemann Hypothesis (RH) building on a commonly applied mathematical framework. The proposed solutions are building on common mathematical solution concepts and tools, e.g. Hilbert scale,
Hilbert (resp. Riesz) transform(s), Hilbert-Polya conjecture. The proposed
mathematical framework is also suggested to build a unified quantum field and gravity field theory, while using the proposed mathematical framework to overcome open mathematical challenges of the D. Loop Quantum Theory (LQT)
In order to prove the The corresponding formulas, when replacing the Gaussian function by its Hilbert transform, are well known: the Hilbert transform of the Gaussian is given by the Dawson integral (GaW). Its properties are e.g. provided in ((AbM) chapter 7, (BrK4) lemma D1). The Dawson function is related to a special Kummer function in a similar form than the (error function) erf(x)-function resp. the li(x)-function ((AbM) (9.13.1), (9.13.3), (9.13.7). A characterization of the Dawson function as an sin-integral (over the positive x-axis) of the Gaussian function is given in ((GrI) 3.896 3.). Its Mellin transform is provided in ((GrI) 7.612, (BrK4) lemma S2). The asymptotics of the zeros of those degenerated hypergeometric functions are given in (SeA) resp. ((BrK4) lemma A4). The fractional part function related Zeta function theory is provided in ((TiE) II). With respect to the considered distributional Hilbert
spaces H(-1/2) and H(-1) we note that the Zeta function is an integral
function
of order 1 and an element of the distributional Hilbert space H(-1). This property is an outcome of the relationship between the Hilbert
spaces above, the Dirichlet series theory (HaG) and the Hardy space
isometry as provided in e.g. ((LaE), §227, Satz 40). With respect to the
physical aspects below we refer to (NaS), where the H(1/2) dual space
of H(-1/2) on the circle (with its inner product defined by a Stieltjes
integral) is considered in the context of Teichmüller theory and the
universal period mapping via quantum calculus. For the corresponding Fourier series analysis we refer to ((ZyA) XIII, 11). The approximation by polynomials in a complex domain leads to several notions and theorems of convergence related to Newton-Gaussian and cardinal series. The latter one are closely connected with certain aspects of the theory of Fourier series and integrals. Under sufficiently strong conditions the cardinal function can be resolved by Fourier's integral. Those conditions can be considerably relaxed by introducing Stieltjes integrals resulting in (C,1) summable series ((WhJ1) theorems 16 & 17, (BrK4) remarks 3.6/3.7). The bounded Hermitian operator definition with appropriately defined domain (but still unbounded in the above Banach spaces due to the Sobolev embedding theorem, see also below)). The corresponding Hilbert scale framwork plays also a key role on the inverse problem for the double layer potential. The corresponding model problem (w/o any compact disturbance operator) with the Newton kernel enjoys a double layer potential integral operator with the eigenvalue 1/2 (EbP).The alternatively proposed Dawson (baseline) function leads to an
alternative entire Zeta function definition The Riemann entire Zeta function Z(s) enjoys the functional equation in the form Z(s)=Z(1-s). The RH is equivalent to the One proof of the Riemann functional equation is based on the chi(s) is defined according to ((TiE) (2.1.12). The Hilbert transform of the fractional part function is given by the log(sin(x))-function. After some calculations (see also (BrK4) lemma 1.4, lemma 3.1 (GrI) 1.441, 3.761 4./9., 8.334, 8.335) the corresponding alternative zeta(*,s) function is given by .zeta(*,1-s) * s = zeta(1-s) * tan(c*s)The above indicates an alternative Gamma function G(*,s/2):=G(s/2)*tan(cs)/s with identical asymptotics for x
--> 0. It is proposed to revisit Riemann's proof of the formula for J(x) and its related fomula for the prime number counting function, resulting into the famous In (TiE) theorem 4.11, an approximation to the In ((BrK4) lemma 3.4, lemma A12/19) the function P(x)
is considered in the context of (appreciated) quasi-asymptotics of
(corresponding) distributions ((ViV) p. 56/57) and the Riemann mapping
theorem resp. the Schwarz lemma. The considered "function" g(x):=-d/dx(cot(x)) (whereby the cot-"function" is an element of H(-1)) is auto-model (or regular varying) of order -1. This condition and its corresponding asymptotics property ((BrK) lemma 3.4) provide the prereqisitions of the In (GrI) 8.334, the relationship between the the cot-
and the Gamma function is provided. From (BeB) 8. Entry17(iii)) we
quote: " With respect to the The The until today not successfull attempts to define a (loop quantum theory (LQT) C. Rovelli) is the choice of a different algebra of basic field functions: a
noncanonical algebra based on the holonomics of the gravitational
connections ((RoC) 1.2.1). The holonomy (or the "Wilson
loop") is the matrix of the parallel transport along a closed curve and spacetime itself is formed by loop-like states.
Therefore the position of a loop state is relevant only with respect to
other loops, and not with respect to the background. The state
space of the theory is a separable Hilbert space spanned by loop states,
admitting an orthogonal basis of spin network states, which are formed
by finite linear combinations of loop states, and are defined precisely
as the spin network states of a lattice Yang-Mills theory." The
proposed distributional quantum state H(-1/2) with corresponding inner product admits and requires
In the following we briefly sketch the conceptual common solution elements motivating our terminology of a "common
Hilbert space framework" to solve the three considered Millenium
problems (and a few other related and considered ones).
The common Hilbert scale is about the Hilbert spaces H(a) with a=1,1/2,0,-1/2,-1 with its corresponding inner products The element of H(-1), but not an element of H(-1/2).Therefore, there is a characterization of the Zeta function on the critcal line in the form ((Z,v)) for all v ex H(0). As the "test space" H(0) is compactly embedded into H(-1/2) this shows that there is an extended Zeta function Z(*)=Z+Z(#) with the characterization ((Z(*),v)) for all v ex H(-1/2), where Z can be interpreted as orthogonal approximation of Z(*) with discrete spectrum. The The Riemann's
"workaround" function h(x) do have an obvious linkage to the
" Conceptually, dealing with the isometric mapping Hilbert transform instead of a second order operator in the form x*P(g(x)) (or the commutator (P,Q)) goes along with a few other opportunities. For example, it enables a correspondingly defined variational representation of the Maxwell equations in a vaccum, whereby its solutions do not need any callibration transforms to ensure wave equation character; therefore, the arbitrarily chosen Lorentz condition for the electromagnetic potential (to ensure Lorentz invariance in wave equations) and its corresponding scalar function ((FeR), 7th lecture) can be avoided. At the same point in time it enables alternative concepts in GRT regarding concepts like current (flexible") metrical affinity, affine connexions and local isometric 3D unit spheres dealing with rigid infinitesimal pieces, being replaced by The newly proposed "fluid/quantum state" Hilbert space H(-1/2) with its closed orthogonal subspace of H(0) goes also along with a combined usage of L(2) waves governing the H(0) Hilbert space and "orthogonal" wavelets governing the H(-1/2)-H(0) space. The wavelet "reproducing" ("duality") formula provides an additional degree of freedom to apply wavelet analysis with appropriately (problem specific) defined wavelets, where the " In SMEP (Standard Model of Elementary Particles) symmetry plays a key role. Conceptually, the SMEP starts with a set of fermions (e.g. the electron in quantum electrodynamics). If a theory is invariant under transformations by a symmetry group one obtains a conservation law and quantum numbers. Gauge symmetries are local symmetries that act differently at each space-time point. They automatically determine the interaction between particles by introducing bosons that mediate the interaction. U(1) (where probability of the wave function (i.e. the complex unit circle numbers) is conserved) describes the elctromagnetic interaction with 1 boson (photon) and 1 quantum number (charge Q). The group SU(2) of complex, unitary (2x2) matrices with determinant I describes the weak force interaction with 3 bosons (W(+), W(-), Z), while the group SU(3) of complex, unitary (3x3) matrices describes the strong force interaction with 8 gluon bosons. With respect to the open Millenium The electromagnetic interaction has gauge invariance
for the probability density and for the Dirac equation. The wave
equation for the gauge bosons, i.e. the A "3D challenge" like the NSE above is also valid,
when solving the monochromatic scattering problem on surfaces of
arbitrary shape applying electric field integral equations. From (IvV)
we recall that the (integral) operators A and A(t): H(-1/2) -->
H(1/2) are bounded Fredholm operators with index zero. The underlying
framework is still the standard one, as the domains are surfaces, only.
An analog approach as above with correspondingly defined surface domain
regularity is proposed.
Replacing the affine connexion and the underlying covariant derivative concept by a geometric structure with corresponding inner product puts the spot on the This conjecture asserts that any compact 3-manifold can be cut in a reasonably canonical way into a union of geometric pieces. In fact, the decomposition does exist. The point of the conjecture is that the pieces should all be geometric.
There are precisely eight homogeneous spaces (X, G) which are needed for
geometric structures on 3-manifolds.
The symmetry group SU(2) of quaternions of absolute value one (the model for the weak nuclear force interaction between an electron and a neutrino) is diffeomorph to S3, the unit sphere in R(4). The latter one is one of the eight geometric manifolds above (ScP). We mention the two other relevant geometries, the Euclidean space E3 and the hyperbolic space H3. It might be that our universe is not an either... or ..., but a combined one, where then the "connection" dots would become some physical interpretation. Looking from an Einstein field equation perspective the Ricci tensor is a second order tensor, which is very much linked to the Poincare conjecture, its solution by Perelman and to S3 (AnM). The The single scalar equation for the Ricci potential
(CaJ) might be interpreted as the counterpart of the The link back to a Hilbert space based theory might be provided by the theory of spaces with an indefinite metric ((DrM), (AzT), (DrM), (VaM)). In case of the L(2) Hilbert space H, this is about a decomposition of H into an orthonal sum of two spaces H1 and H2 with corresponding projection operators P1 and P2 relates to the concepts which appear in the problem of S. L. Sobolev concerning In an universe model with appropriately connected geometric manifolds the corresponding symmetries breakdowns at those "connection dots" would govern corresponding different conservation laws in both of the two connected manifolds. The Noether theorem provides the corresponding mathematical concept (symmetry --> conservation laws; energy conservation in GT, symmetries in particle physics, global and gauge symmetries, exact and broken). Those symmetries are associated with "non-observables". Currently applied symmetries are described by finite- (rotation group, Lorentz group, ...) and by infinite-dimensional (gauged U(1), gauged SU(3), diffeomorphisms of GR, general coordinate invariance...) Lie groups. A manifold geometry is defined as a pair (X,G), where X
is a manifold and G acts transitively on X with compact point
stabilisers (ScP). Related to the key tool "Hilbert transform" resp.
"conjugate functions" of this page we recall from (ScP), that Kulkarni
(unpublished) has carried out a finer classification in which one
considers pairs (G,H) where G is a Lie group, H is a compact subgroup
and G/H is a simple connected 3-manifold and pairs (G1,H1) and (G2,H2)
are equivalent if there is an isomorphism G1 --> G2 sending H1 to a
conjugate of H2. Thus for example, the geometry S3 arises from three
distinct such pairs, (S3,e), (U(2),SO(2)), (SO(4),SO(3)). Another
example is given by the Bianchi classification consisting of all simply
connected 3-dimensional Lie groups up to an isomorphism.
The central mathematical concepts of the GRT are differentiable manifolds, affine connexions with the underlying covariant derivative definition on corrresponding tangential (linear) vector spaces. Already the "differentiability" condition is w/o any physical justification. The only "affine" connexion concept and its corresponding locally defined metrical space framework jeopardizes a truly infinitesimal geometry, which is compatible with the Hilbert space framework of the quantum theory and the proposed distributional Hilbert space concept in (BrK). In sync with the above we propose a generalized Gateaux differential operator: let H(1/2) = H(1) + H(*) denote the orthogonal decomposition of the alternatively proposed "energy/momentum/velocity" Hilbert space, whereby H(1) denotes the (compactly embedded) standard energy space with its inner product, the Dirichlet integral; "lim" denotes the limes for t --> 0 for real t. Then for x,y ex H(1/2) the operator VF(x,y) is defined by VF(x,y):=lim((F(x+t*y)-F(x))/t), whereby the limes is understood in a weak H(-1/2) sense. The operator is homogeneous in y; however, it is not always a linear operator in y ((VaM) 3.1). The main tools used in geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing space-time. A Lorentz manifold L is likewise equipped with a metric tensor g, which is a nondegerated symmetric bilinear form on the tangential space at each point p of L. The Minkowski metric is the metric tensor of the (flat space-time) Minkowski space. The least action principle can refer to the family of variational principles. The most popular among these is Hamilton's principle of least action. It states that the action is stationary under all path variations q(t) that vanishes at the end points of the path. It does not not strictly imply a minimization of the action. The Offenbar haben wir beim Energieprinzip eine typische Entwicklung vor uns: wenn das Prinzip der reflektierenden Urteilskraft mit einer seiner Maximen vollen Erfolg gehabt hat, rückt sein Ergebnis aus dem Reich der Vernunft im Kantischen Sinne, zu welchem die reflektierende Urteilskraft gehört, in die Sphäre des Verstandes herab und wird zum allgemeinen Naturgesetz (".law of nature)The The Einstein-Hilbert functional is an ) plus a divergence integral, that is an integral whose integrand is of the form div(" This is where a alternative field-action functional of gravitation in a alternative framework (as proposed above) can be defined, based on a w). Hence for the corresponding variations of theh potential functions g(i,k) the variations of both funtionals are identical; therefore the replacement of the physically required scalar density G by the integrand of the W(g) is justified (as the essential feature of the Hamilton's principle is fulfilled with W(g))." in a "scalar density" functionPlemelj" (Stieltjes integral) sense.The electromagnetic field is built up from the co-efficients of an invariant - replacing of the mathematical " - replacing the manifold concept by a (semi) Hilbert
space-based concept, where a non-linear invariant integral functional F(V(g)) is defined by a
distributional (semi-) inner product, which is equivalent to a corresponding
functional F(R(g)) of a related inner product (where R denotes the Riesz
operators (which commute with translations & homothesis having nice
properties relative to rotations)) plus a (non-linear) compact disturbance term; the concept enables variational methods of nonlinear operators based on Stieltjes and curvilinear integrals (VaM). The Yang-Mills functional is of similar structure than the Maxwell functional regarding the underlying constant fundamental tensor. The field has the property of being self-interacting and equations of motions that one obtains are said to be semilinear, as nonlinearities are both with and without derivatives. The To merge two inconstent theories requires changes on both sides. In the above case this is about a newly proposed common "mass/substance element" concept, alternatively to the today's "mass density" concept, while, at the same time, the linear algebra tensor tool (e.g. a "density" tensor) describing classical PDE systems is replaced by non-linear operator equations defined by weak (variational) functional systems. Those (weak) equations provides the mathematical model of physical phenomena, while its correspondin classical PDE systems (requiring purely mathematical additional regularity assumptions) are interpreted as approximation solutions, only.
As a shortcut reference to geometrodynamics is
given by (WhJ). For a review of discoveries in the nonlinear dynamics of curved spacetime, we refer to ((ScM). An introduction to the foundations and tests of
gravitation and geometrodynamics or the meaning and origin of inertia in
Einstein theory is provided in (CiI). In ((CiI) 4.6) the Gödel model universe is discussed, which is a four-dimensional model universe, homogeneous both in space and time, which admits the whole four-dimensional simply transitive group of isometries, in other words, a space-time that admits all four "simple translations" as independent Killing vectors. As the Gödel model universe is homogeneous both in space and time it is stationary. In other words, in this model the cosmological fluid is characterized by zero expansion and zero shear. Thus the Gödel model runs into difficulty with the expansion of the universe. " The related clarifcations regarding the distortion tensor or gravitomagnetic field is provided in ((CiI) §5.2.6, § 5.2.7). The Laplacian equation for the gravitomagnetic vector
potential W, in terms of the current J of mass-energy is discussed in
((CiI) 5.3). The Neumann problem and its related integral equations with double layer potential leads to the Prandtl operator, defining a well posed integral equation in case of domain H(1/2) with range H(-1/2) ((LiI) theorem 4.3.2). whereby (((*,*))) defines the H(-1) inner product and
((*,*)) defines the H(-1/2) inner product of the corresponding Hilbert
scales building on the eigen-pair solutions of the Prandtl operator
equation with domain H(1/2). The proposed alternative Hilbert space based framework provides
also a "variational wave equation/ function" based approach of the "evolution of
geometric structures on 3-manifolds" in the context of Thurston's
" "T If we think these ideas consistently through to the end we must expect the whole inertia, that is, the whole g(i,k)-field, to be determined by the matter of the universe, and not mainly by the boundary conditions at infinity.
The wave equation can be derived from the Maxwell equations by applying the rot-operator. It results into the "light" phenomenon. A similar tranformation is not possible for Einstein equations, which results into the "gravitation" phnomenon. The "approximation" approach is about the split g(i,k)=m(i,k)+h(i,k), where m(i,k) denotes the flat Minkowski metric. The perturbance term h(i,k) admits a retarded (only) potential representation, representing a gravitational perturbance propagating at the speed of light ((CiI) 2.10). An alternative splitting with defined distortion tensor enabling an analogue approach with electrodynamics is provided in ((CiI) 5.2.7). In ((CiI) (2.7.10)) an „ effective energy-momentum tensor for matter, fields and gravity field". As the Einstein (gravity) tensor is derived from the condition of a divergence-free energy-momentum tensor, this results to an alternative Einstein tensor. The additional term of this alternative Einstein tensor could be interpreted as "cosmologic term", not to ensure a static state of the universe (which is not the case due to Hubbles observations), but to model the "vacuum energy" properly. This then would also be in sync with the physical interpretation of the corresponding term in the modified Maxwell equations with its underlying split of divergence-free and rotation-free tensors. At the same point in time the approach avoids the affine connexion concept and the "differentiable" manifolds regularity requirement, which is w/o any physical justification.There are eight 3-dimensional geometries in the context of "nice"
metrics. The nicest metrics are those with a constant curvature, but
there are other ones. Their classification in dimension three is due to
Thurston (ScP). In (GrJ) philosophical aspects of the geometrodynamics are considered. We quote from the cover letter summary: The above
In (CoR) there is a
We mention that the existing
electromagnetic phenomena on earth are the result of plasma physics phenomena
underneath the earth crust. Those “activities” are all triggered by gravitation "forces". The above (distributional) Hilbert space based
alternative geometrodynamic modelling framework provides an alternative
approach to " ((RoC) 1.2.1). The holonomy (or the "Wilson loop") is the matrix of the parallel transport along a closed curve. ... In LQT, the holonomy becomes a quantum operator that creates "loop states" (to overcome the issue of current dynamics model of coupled gravity + matter system, simply defined by adding the terms defining the matter dynamics to the gravitational relativistic hamiltonian ((RoC) 7.3)). ..." The proposed distributional quantum state H(-1/2) above admits and requires infinite linear combinations of those "loop states" (which we call "quantum fluid" state), i.e. overcomes the Spacetime itself is formed by loop-like states. Therefore the position of a loop state is relevant only with respect to other loops, and not with respect to the background. ... The state space of the theory is a separable Hilbert space spanned by loop states, admitting an orthogonal basis of spin network states, which are formed by finite linear combinations of loop states, and are defined precisely as the spin network states of a lattice Yang-Mills theory.current challenge of LQT defining the scalar product of the spin network state Hilbert space ((RoC) 7.2.3). The physical space is a quantum superposition of "" in LQT corresponds to an orthogonal projection of H(-1/2) onto H(0). This othogonal projection can be interpreted as a general model for a "spin networksspontaneous symmetry break down".
axiom of observables (each
physical observable is represented as a linear Hermitian operator
of
the state Hilbert space). The corresponding mathematical model and its
solutions are governed by the Heisenberg uncertainty inequality. As the
observable space needs to support statistical analysis the
Hilbert space, this Hilbert space needs to be at
least a subspace of A.
At the same point in time, if plasma is considered as
sufficiently collisional, then it can be well-described by fluid-mechanical
equations. There is a hierarchy of such hydrodynamic models, where the magnetic
field lines (or magneto-vortex lines) at the limit of infinite conductivity is
“frozen-in” to the plasma. The “mother of all hydrodynamic models is the Hcontinuity equation treating observations
with macroscopic character, where fluids and gases are considered as continua.
The corresponding infinitesimal volume “element” is a volume, which is small
compared to the considered overall (volume) space, and large compared to the
distances of the molecules. The displacement of such a volume (a fluid
particle) then is a not a displacement of a molecule, but the whole volume
element containing multiple molecules, whereby in hydrodynamics this fluid is
interpreted as a mathematical point.The common distributional Hilbert space framework is also proposed for a proof of the Landau damping alternatively to the approach from C. Villani. Our approach basically replaces an analysis of the classical (strong) partial differential (Vlasov) equation (PDE) in a corresponding Banach space framework by a quantum field theory adequate (weak) variational representation of the concerned PDE system. This goes along with a corresponding replacement of the “hybrid” and “gliding” analytical norms (taking into account the transfer of regularity to small velocity scales) by problem adequate Hilbert space norms H(-1/2) resp. H(1/2). The latter ones enable a "fermions quantum state" Hilbert space H(0), which is dense in H(-1/2) with respect to the H(-1/2) norm, and its related (orthogonal) "bosons quantum state" Hilbert space H(-1/2)-H(0), which is a closed subspace of H(-1/2).
The
The In fluid description of
plasmas ( The
When one wants to treat
the time-harmonic Maxwell equations with variational methods, one has to face
the problem that the natural bilinear form is not coercive on the whole Sobolev
space.
On can, however, make it coercive by adding a certain bilinear form on the
boundary of the domain (vanishing on a subspace of H(1)), which causes a change in the natural
boundary conditions.
The mathematical tool to
distinguish between unperturbed cold and hot plasma is about the Debye length
and Debye sphere. The corresponding interaction (Coulomb) potential of
the
As a shortcut reference to the underlying mathematical principles of classical fluid mechanics we refer to(SeJ).
(AnM) Anderson M. T., Geometrization of 3-manifolds via the Ricci flow, Notices Amer. Math. Sco. 51, (2004) 184-193 (AzT) Azizov T. Y., Ginsburg Y. P., Langer H., On Krein's papers in the theory of spaces with an indefinite metric, Ukrainian Mathematical Journal, Vol. 46, No 1-2, 1994, 3-14 (BeB) Berndt B. C., Ramanujan's Notebooks, Part I, Springer Verlag, New York, Berlin, Heidelberg, Tokyo, 1985 (BiP) Biane P., Pitman J., Yor M., Probability laws related to the Jacobi theta and Riemann Zeta functions, and Brownian excursion, Amer. Math. soc., Vol 38, No 4, 435-465, 2001 (BrK) Braun K., A new ground state energy model, www.quantum-gravitation.de (BrK2) Braun K., Global existence and uniqueness of 3D Navier-Stokes equations (BrK3) Braun K., Some remarkable Pseudo-Differential Operators of order -1, 0, 1 (BrK4) Braun K., A Kummer function based Zeta function theory to prove the Riemann Hypothesis and the Goldbach conjecture (BrK5) An alternative trigonometric integral representation of the Zeta function on the critical line (BrK related papers) www.navier-stokes-equations.com/author-s-papers (CaD) Cardon D., Convolution operators and zeros of entire functions, Proc. Amer. Math. Soc., 130, 6 (2002) 1725-1734 (CaJ) Cao J., DeTurck D., The Ricci Curvature with Rotational Symmetry, American Journal of Mathematics 116, (1994), 219-241 (ChK) Chandrasekharan K., Elliptic Functions, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985 (CiI) Ciufolini I., Wheeler J. A.,
Gravitation and Inertia, Princeton University Press , Princeton, New Jersey,
1995 (CoR) Courant R., Hilbert
D., Methoden der Mathematischen Physik II, Springer-Verlag, Berlin, Heidelberg,
New York, 1968 (EbP) Ebenfelt P., Khavinson D., Shapiro H. S., An inverse problem for the double layer potential, Computational Methods and Function Theory, Vol. 1, No. 2, 387-401, 2001 (ElL) Elaissaoui L., El-Abidine Guennoun Z., Relating log-tangent integrals with the Riemann zeta function, arXiv, May 2018 (EsR) Estrada R., Kanwal R. P., Asymptotic Analysis: A Distributional Approach, Birkhäuser, Boston, Basel, Berlin, 1994 (FaK) Fan K., Invariant subspaces of certain linear operators, Bull. Amer. Math. Soc. 69 (1963), no. 6, 773-777 (FeR) Feynman R. P., Quantum Electrodynamics, Benjamin/Cummings Publishing Company, Menlo Park, California, 1961 (GaW) Gautschi W., Waldvogel J., Computing the Hilbert Trnasform of the Generalized Laguerre and Hermite Weight Functions, BIT Numerical Mathematics, Vol 41, Issue 3, pp. 490-503, 2001 (GrI) Gradshteyn I. S., Ryzhik I. M., Table of
integrals series and products, Academic Press, New York, San Franscisco,
London, 1965 (GrJ) Graves J. C., The conceptual foundations
of contemporary relativity theory, MIT Press, Cambridge, Massachusetts, 1971 (HaG) Hardy G. H., Riesz M., The general theory of Dirichlet's series, Cambridge University Press, Cambridge, 1915 (HoM) Hohlschneider M., Wavelets, An Analysis Tool, Clarendon Press, Oxford, 1995 (HoA) Horvath A. G., Semi-indefinite-inner-product and generalized Minkowski spaces, arXiv (IvV) Ivakhnenko, V. I., Smirnow Yu. G., Tyrtyshnikov
E. E., The electric field integral equation: theory and algorithms,
Inst. Numer. Math. Russian of Academy Sciences, Moscow, Russia (KaM) Kaku M., Introduction to Superstrings and M-Theory, Springer-Verlag, New York, Inc., 1988 (KeL) Keiper J. B., Power series expansions of Riemann's Zeta function, Math. Comp. Vol 58, No 198, (1992) 765-773 (KlB) Kleiner B., Lott J., Notes on Perelman s papers, Mathematics ArXiv (KnA) Kneser A., Das Prinzip der kleinsten Wirkung von Leibniz bis zur Gegenwart, B. G. Teubner, Leipzig, Berlin, 1928 (LaE) Landau E., Die Lehre von der Verteilung der Primzahlen I, II, Teubner Verlag, Leipzig Berlin, 1909 (LeN) Lebedev N. N., Special Functions and their Applications, translated by R. A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, New York, 1965 (LiI) Lifanov I. K., Poltavskii L. N., Vainikko G. M., Hypersingular Integral Equations and their Applications, Chapman & Hall/CRC, Boca Raton, London, New York, Washington, D. C., 2004
(LiP) Lions
P. L., On Boltzmann and Landau equations, Phil. Trans. R. Soc. Lond. A, 346,
191-204, 1994 (LiP1)
Lions P. L., Compactness in Boltzmann’s equation via Fourier integral operators
and applications. III, J.
Math. Kyoto Univ., 34-3, 539-584, 1994
(LiX) Li Xian-Jin, The Positivity of a Sequence of Numbers and the Riemann Hypothesis, Journal of Number Theory, 65, 325-333 (1997) (LoA) Lifanov I. K., Poltavskii L. N., Vainikko G. M.,
Hypersingular Integral Equations and Their Applications, Chapman &
Hall/CRC, Boca Raton, London, New York, Washington, D. C. 2004 (MoJ) Morgan J. W., Tian G., Ricci Flow and the Poincare Conjecture, Mathematics ArXiv (NaS) Nag S., Sullivan D., Teichmüller theory and the universal period mapping via quantum calculus and the H space on the circle, Osaka J. Math., 32, 1-34, 1995 (PeR) Penrose R., Cycles of Time, Vintage, London, 2011 (PoG) Polya G., Über Nullstellen gewisser ganzer Funktionen, Math. Z. 2 (1918) 352-383 (RoC) Rovelli C., Quantum Gravity, Cambridge University Press, Cambridge, 2004 (ScP) Scott P., The Geometries of 3-Manifolds (SeA) Sedletskii A. M., Asymptotics of the Zeros of Degenerated Hypergeometric Functions, Mathematical Notes, Vol. 82, No. 2, 229-237, 2007 (SeJ) Serrin J., Mathematical Principles of Classical Fluid Mechanics (ShM) Scheel M. A., Thorne K. S., Geodynamics, The Nonlinear Dynamics of Curved Spacetime (StE) Stein E. M., Conjugate harmonic functions in several
variables (ThW) Thurston W. P., Three Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry, Bulletin American Mathmematical society, Vol 6, No 3, 1982 (TiE) Titchmarsh E. C., The theory of the Riemann Zeta-function, Clarendon Press, London, Oxford, 1986 (VaM) Vainberg M. M., Variational Methods for the
Study of Nonlinear Operators, Holden-Day, Inc., San Francisco, London,
Amsterdam, 1964 (VeW) Velte W., Direkte Methoden der Variationsrechnung, B. G. Teubner, Stuttgart, 1976 (VlV) Vladimirow V. S., Drozzinov Yu. N., Zavialov B. I., Tauberian Theorems for Generalized Functions, Kluwer Academic Publishers, Dordrecht, Boston, London, 1988 (WhJ) Wheeler J. A., On the Nature of Quantum Geometrodynamics (WhJ1) Whittaker J. M., Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935 (WeH) Weyl H., Space, Time, Matter, Cosimo Classics, New York, 2010 (WeH1) Weyl H., Matter, structure of the world, principle of action, ...., in (WeH) §34 ff. (WeH2) Weyl H., Was ist Materie? Verlag Julius Springer, Berlin, 1924 (ZhB) Zhechev B., Hilbert Transform Relations (ZyA) Zygmund A., Trigonometric series, Volume I & II, Cambridge University Press, 1959 | ||||||||||||||||