Quantum world versus creation. Igor V Bayak. It is well ..... in the simplest case closed in a circle, and the question of classification of topological features and ...
Quantum world versus creation Igor V Bayak It is well known [?] that matter (as a philosophical category) in the case of its simplest motion has a mechanical (physical) form. However, there is no mathematical description of the moving matter from which the formation of this form of matter would follow. At the same time, this monograph, which is a collection of mathematical notes of the author, is designed to ll this gap. To achieve this goal, we need to choose an adequate mathematical design of moving matter and correctly answer the question of where and how it moves. In the Preface, on the example of a ow moving along the surface of an in nite cylinder, we constructed a simple model of classical (not quantum) mechanics, and in the introduction we focus on the quantum aspects of our model. First of all, we consider the algebra of vector elds corresponding to the local algebraic structure of vacuum. In this case, we assume that the local algebraic structure of the vacuum is embedded in the algebraic structure of the equations of motion of a quantum particle. Thus, we are interested in vector- eld representations of alternion algebras [?] formed by Pauli and Dirac matrices, i.e. Cliord algebras in which Euclidean space and Minkowski space are embedded respectively. Since we are interested in nite-dimensional algebras are represented by complex matrices, we begin with a vector- eld representation of the algebra of complex numbers. Let the Cartesian plane be the product of two smooth vector elds X ? Y = ∇X Y , where ∇ is the derivative of the vector eld Y in the direction X . Then, for linear vector elds
I = y∂x − x∂y
E = x∂x + y∂y ,
1
(0.1)
relations are executed
E 2 = E ? E = ∇E E = x∂x + y∂y = E, E ? I = I ? E = ∇E I = ∇I E = y∂x − x∂y = I, I 2 = I ? I = ∇I I = −x∂x − y∂y = −E.
(0.2) (0.3) (0.4)
Hence, the vector eld I serves as a generator of the algebra of linear vector elds, which is isomorphic to the algebra of complex numbers h(I, ∗)iR ∼ = C. Note that the current lines of the vector eld corresponding to the complex number a + b i, on the Cartesian plane (x1 , x2 ) have the form of spirals, which are given by the parametric equation
x2 (τ ) = eaτ +z0 sin(bτ + ϕ0 )
x1 (τ ) = eaτ +z0 cos(bτ + ϕ0 ),
(0.5)
where x1 (0) = ez0 cos ϕ0 and x2 (0) = ez0 sin ϕ0 , and on the cylinder (z, ϕ) they are helical lines
z(τ ) = aτ + z0 ,
ϕ(τ ) = bτ + ϕ0
(0.6)
where z(τ ˙ ) = a, ϕ(τ ˙ ) = b. Thus, from the linear vector eld on the plane we obtained a constant vector eld on the cylinder associated with the vacuum ow of matter. And since the area swept by the ring, which is carried away by the vacuum ow of matter, is equal to S = z(τ )ϕ(τ ), then according to the principle of ow minimality the motion of the ring will be invariant with respect to such ow transformations that z˙0 (τ )ϕ˙ 0 (τ ) = const. In other words, the covering coordinates of the cylinder (z, ϕ) in describing the free motion of the ring should be considered as the isotropic coordinates of the pseudo-Euclidean plane (x, t), where
z = t − x, ϕ = t + x, Then
2t = z + ϕ 2x = ϕ − z
2 2 z˙0 (τ )ϕ˙ 0 (τ ) = t˙0 (τ ) − x˙0 (τ ) = const
(0.7)
(0.8)
Moreover, by winding the second isotropic coordinate on the circle z˜ = |z| mod 2π , we transform the cylinder into a torus and take the rst step towards the simulation of quantum mechanics. Indeed, since the area swept by the ring on the torus is now quantized, we can talk about the covering of the torus area as the proper rotation of the ring on the torus ˙ ψ(τ ) = e2πiS = e2πiz˙ ϕτ = e2πi(t 2
2
˙2 −x˙ 2 )τ 2
(0.9)
So, we have theoretical grounds for modeling one-dimensional quantum mechanics. And since we would not like to limit ourselves to a smalldimensional model, we will now try to expand its dimension to the necessary one. Taking into account that the quantum particle is described by a 4component Dirac spinor, it is worth trying to match it with a direct product of 4 linear vector elds, each of which is isomorphic to the corresponding complex number. In order to place these vector elds, we need four Cartesian planes (x1 , x2 ), (x3 , x4 ), (x5 , x6 ), (x7 , x8 ), each of which contains the corresponding component of the linear vector eld of the 8-dimensional Cartesian space R8 . At the same time, in order to study the symmetries of this vector eld, we should remember that Dirac's matrix algebra acts on Dirac spinors, which is is isomorphic to M4 (C). In the rst article of this collection it is shown that the matrix algebra M4 (C) of interest to us can be represented by the algebra of linear vector elds, in which the generating vector elds (algebra generators) are tangent to the hypersphere of 8-dimensional Euclidean space and pseudo-Euclidean spaces with a neutral metric, namely
x21 + x22 + x23 + x24 + x25 + x26 + x27 + x28 = const, x21 + x22 − x23 − x24 + x25 + x26 − x27 − x28 = const, x21 + x22 + x23 + x24 − x25 − x26 − x27 − x28 = const, x21 + x22 − x23 − x24 − x25 − x26 + x27 + x28 = const
(0.10)
Knowledge of the algebraic structure of vacuum can be used to model the shape of moving matter. Indeed, if the matter moves on the surface of the seven-dimensional sphere S 7 , then given the fact that the intersection of the 1 sphere x21 + x22 + x23 + x24 + x25 + x26 + x27 + x28 = e τ , where τ evolutionary argument, and, for example, the pseudosphere x21 + x22 + x23 + x24 − x25 − x26 − 1 x27 − x28 = e− τ is the product of spheres S 3 × S 3 1
1
e τ + e− τ + + + = , 2 1 1 e τ − e− τ 2 2 2 2 x5 + x 6 + x7 + x8 = 2
x21
x22
x23
x24
(0.11)
it can be assumed that the foliation of a vacuum vector eld at a certain moment of evolution and without taking into account the evolutionary component of this eld has the form of a product of spheres. 3
Let us now turn to the question of the coordinate representation of a cylindrical (toroidal) variety formed by the current lines of eight linearly independent vacuum vector elds. First of all, the doublet of Minkowski spaces, namely the space (x, y, z, t) with quadratic metric of the form t2 − x2 − y 2 − z 2 and the space (t∗ , x∗ , y ∗ , z ∗ ) with quadratic metric of the form x∗2 + y ∗2 + z ∗2 − t∗2 . Then form from them the 8-dimensional Finslerian the product of (x, y, z, t∗ , t, x∗ , y ∗ , z ∗ ) with quadratic metric of the form x∗2 + y ∗2 + z ∗2 − t∗2 + t2 − x2 − y 2 − z 2 . If we now collapse the pseudoEuclidean planes (x, x∗ ), (y, y ∗ ), (z, z ∗ ), (t, t∗ ) into a cylindrical variety so that the isotropic coordinates x + x∗ , y + y ∗ , z + z ∗ , t + t∗ are wound on the de ning circles of the corresponding cylinders, then the coordinates of the collapsed 8-dimensional space become helical. Similarly, if all isotropic coordinates are wound on the given circles of the corresponding tori, then we obtain a toroidal variety in which the coordinates (x, y, z, t∗ , t, x∗ , y ∗ , z ∗ ) also have a helical, but sometimes also a closed form. At the same time, in our minimized Finsler metric space, we should set the vector eld ξ(x) corresponding to the ow of moving matter, which in the vacuum case is given by a constant time-like vector ξ(x) = c. An arbitrary vector eld ξ(x) forms a certain hyperbolic angle with the vacuum vector c, which serves as a measure of the deviation of the vector eld ξ(x) from the vacuum state. In the case where the hyperbolic angle tends to in nity, the vector eld has a topological feature because it becomes isotropic and its current lines are closed in a circle. In turn, since the topological feature of a vector eld is a ring, this ring, when fully rotated in a toroidal variety, sweeps it completely, and therefore has such a characteristic as the angular velocity ω = (ωt , ωt∗ , ωx , ωx∗ , ωy , ωy∗ , ωz , ωz∗ ), a scalar product
S = ω · x = ωt t − ωt∗ t∗ − ωx x + ωx∗ x∗ − ωy y + ωy∗ y ∗ − ωz z + ωz∗ z ∗ (0.12) measures at the point x = (x, y, z, t∗ , t, x∗ , y ∗ , z ∗ ) the phase action (eigenvalue) of an absolutely non-localized singularity. However, in the case when, due to the small diameter of the winding circles of isotropic lines with an asterisk, the space of the global observer is limited by a system of equations
T ∗ = t + t∗ = 0, Y ∗ = y + y ∗ = 0,
X ∗ = x + x∗ = 0, Z ∗ = z + z∗ = 0
(0.13)
there is a classical limit, according to which only the linear coordinates of the 4
cylinders (corresponding coordinates of the tori) are available to the observer
T = t − t∗ = 2t, Y = y − y ∗ = 2y,
X = x − x∗ = 2x, Z = z − z ∗ = 2z
(0.14)
bound by the Minkowski space metric (t, x, y, z). Hence, in the classical limit, the formula for the phase action of a non-localized singularity is simpli ed to a scalar product S = Ωt t − Ωx x − Ωy y − Ωz z (0.15) where Ωt = ωt + ωt∗ , Ωx = ωx + ωx∗ , Ωy = ωy + ωy∗ , Ωz = ωz + ωz∗ . Now, to match our model with classical physics, it is necessary to localize the singularity, and the angular velocity vector of the singularity Ω is to match the 4-momentum of a relativistic particle p according to the formula (0.16)
p = ~Ω Then, in the non-relativistic case, the formulas are true
p = mv = ~k,
E = ~ω
(0.17)
where k = (Ωx , Ωy , Ωz ), ω = Ωt . Absolutely not localized feature serves as a brick for the construction of the mathematical apparatus of quantum mechanics. Indeed, let absolutely non-localized singularity be described by the wave function ΨΩ = eiS , where S = Ωx = Ωt t − Ωx x − Ωy y − Ωz z classical phase action of non-localized singularity. Then, the probabilistic superposition of two nonlocalized singularities can be described by the square of the wave function
Ψ2 = c1 c¯1 ΨΩ1 + c2 c¯2 ΨΩ2
(0.18)
where c1 , c2 ∈ C, and since the squares of the modules of these complex coe cients set the probabilities of observing the corresponding random events, the square of the wave function makes sense of mathematical expectation. However, if we consider the existence of localized features with a shift of the phase steps ΨΩ,ϕ = eiΩx+iϕ = eiΩx eiϕ , it is possible to "remove the root"of the square of the wave function Ψ2 = Ψ∗ Ψ, where 1 2 Ψ = c1 ΨΩ1 + c2 ΨΩ2 = |c1 |eiΩ x+i arg(c1 ) + |c2 |eiΩ x+i arg(c2 )
5
(0.19)
1 2 Ψ∗ = c¯1 ΨΩ1 + c¯2 ΨΩ2 = |¯ c1 |eiΩ x+i arg(¯c1 ) + |¯ c2 |eiΩ x+i arg(¯c2 )
(0.20)
Indeed, since a joint random event consisting of non-localized singularities 1 1 1 eiΩ x+i arg(c1 ) and eiΩ x+i arg(¯c1 ) there is a non-localized singularity eiΩ x , respectively a joint random event, consisting of localized features of the 2 2 2 eiΩ x+i arg(c2 ) and eiΩ x+i arg(¯c2 ) is not a localized feature of the eiΩ x , then by multiplying the appropriate probability of actually random events, we get a convolution of wave functions 1 2 Ψ2 = Ψ∗ Ψ = |c1 ||¯ c1 |eiΩ x + |c2 ||¯ c2 |eiΩ x
(0.21)
Thus, the wave function Ψ should be interpreted as a superposition of nonlocalized singularities equal to the square root of the expectation. On the other hand, the wave function Ψ(x) can be interpreted as the probability amplitude of localization of a singularity, where Ψ∗ (x)Ψ(x) probability of localization of a singularity with zero phase at x, and arg(Ψ(x) phase of localized singularity. It is clear that from the superposition of two nonlocalized singularities we can proceed to the nite, countable and even continuous case Z ψ(x) = c(k)eikx dk (0.22) By simplifying the continuous wave function to one dimension and applying the integral Fourier transform to it, we obtain the wave function Z ψ(k) = c(x)eikx dx (0.23) which tells us that the space of localized singularities is dual to the space of non-localized singularities. Thus, localized features can also serve as building blocks for the mathematical apparatus of quantum mechanics. As an illustration of our analogies to quantum physics, we now show that, in the classical limit, the Markov process of a random walk of a localized singularity leads to a Feynman formulation of quantum mechanics. Let the probabilistic behavior of a localized singularity be described by the Markov random walk process in which the elementary random event is free run. With free path, we associate such random variables as the absolute free path time ∆τ , the free path length ∆x in the Euclidean observer space, and the free ∆x , ϕ initial phase localized path phase length ∆φ = k∆x+ϕ, where k = m ~ ∆τ 6
feature. Then, referring to the probability of free runs are localized features, and changing, for convenience, denote the free path length ∆x on y , without taking into account selection of the multiplier will get the integral amount
Z+∞ my 2 ψ(x, τ0 → τ0 + ∆τ ) = ei ~∆τ +iϕ(x−y) ψ(x − y, τ0 )dy
(0.24)
−∞
which calculates the amplitude of the singularity localization at the point x at the nite free path time τ0 +∆τ by the values of the probability amplitude of the singularity localization at the points x − y at the initial free path time τ0 . The probability amplitude of the feature localization for two free runs is calculated as a product of two integral sums
ψ(x, τ0 → τ0 + ∆τ1 ) × ψ(x, τ0 + ∆τ1 → τ0 + ∆τ1 + ∆τ2 ) = Z+∞ 2 i my +iϕ(x−y) ψ(x − y, τ0 )dy× e ~∆τ1 −∞
Z+∞ 2 i my +iϕ(x−y) ψ(x − y, τ0 → τ0 + ∆τ1 )dy (0.25) e ~∆τ2 −∞
In turn, the probability amplitude of localization of a singularity for n free runs is calculated as the product of n integral sums n Y
ψ(x, τ0 + · · · + ∆τi−1 → τ0 + · · · + ∆τi−1 + ∆τi ) =
i=1
ψ(x, τ0 → τ0 + ∆τ1 ) × · · · × ψ(x, τ0 + · · · + ∆τn−1 → τ0 + · · · + ∆τn−1 + ∆τn ) (0.26) Thus, the Feynman integral over trajectories is essentially the limit of products
lim
∆τi →0 n→∞
n Y
ψ(x, τ0 + · · · + ∆τi−1 → τ0 + · · · + ∆τi−1 + ∆τi )
(0.27)
i=1
and so, keeping in mind the representation of integral sums by the limit of nite sums, the continuum integral can be represented as the in nite limit of 7
the nite product of nite sums. Indeed, suppose on the topological feature of the semiclassical condition is imposed restricting the speed of mean free path from a point xj in the point xk from τi−1 to τi the inequality |xk − xj | ≤ c|τi − τi−1 |. Then, when calculating the probability amplitude of localization of the singularity for N of random free runs, we take into account the nite volume of summation and for the continuum integral we obtain the formula
lim
N →∞ M →∞
M a
ψ(τN , xk ) =
k=1
lim
|τi −τi−1 |→0 |xj −xj−1 |→0
N a M Y
ψ(τi , xk ) =
i=0 k=1 N a M X M Y
F ((τi − τi−1 ), (xk − xj ))ψ(τi−1 , xj ) (0.28)
i=0 k=1 j=1
where F ((τi − τi−1 ), (xk − xj ) is the probability amplitude of the transition from xj to xk from τI−1 to τi . It should also be noted that our model also has a geometric representation that occurs when describing the motion of matter from the point of view of a local observer. Indeed, let the space (x, y, z, t∗ , t, x∗ , y ∗ , z ∗ ) have an arbitrary holonomic vector velocity eld of matter particles ξ , whose current lines serve as the time coordinates t0 of the local observer. We parametrize the time coordinate t0 by the eigentime of the local observer τ equal to one isotropic coordinate of the plane (t, t0 ), and the spatial coordinates are parametrized by the eigenvalue length of the local observer l equal to the other isotropic coordinate of the plane (t, t0 ). Then the curvilinear coordinates (t0 , x0 , y 0 , z 0 ) form a pseudo-Riemannian manifold with a metric dierential form
ds2 = e2ϕ+2ϕx +2ϕy +2ϕz dt2 − e−2ϕ−2ϕx dx2 − e−2ϕ−2ϕy dy 2 − e−2ϕ−2ϕz dz 2 (0.29) which is induced by the vector eld ξ , since ϕ hyperbolic angle between ξ and t∗ , ϕx hyperbolic angle between ξ and x, ϕy hyperbolic angle between ξ and y, ϕz hyperbolic angle between ξ and z. However, all dimensions are not available to the local observer, so we will return to the point of view of the global observer. Let the geometry (shape) of the vector eld ξ(x) be given by the variational equation Z δ ξ ∗ (x)dx = 0 (0.30) l
8
where ξ ∗ (x) covector eld, ξ ∗ (x)dx scalar product of the covector eld at point x on the path dierential at this point, l arbitrary path in 8dimensional space with neutral metric. Then, since the necessary conditions for this equation delivers a dierential equation
d ? n(x) = 0
(0.31)
ξ(x) where n(x) = |ξ( x)| , d external dierential, ? Hodge operator, the geometry of the vector eld ξ(x) is such that its orthogonal 7-dimensional hypersurfaces are locally minimal. In this case, the orthogonal hyperplane globally minimal vector eld we map to vacuum, and locally minimal vector elds together with the singularities in which it is tangent to the isotropic hypersphere we map to elementary particles. No less interesting consequence of the hypothesis of the motion of matter on the surface of the 7-sphere is the existence of the group structure of this motion. On the surface of the 7-sphere there is an in nite-dimensional lie algebra of tangent vector elds, however, since we do not yet have explicitly any dierential form of matter motion on the surface of the 7-sphere, we can not talk about any symmetries of this motion. At the same time, in the 8-dimensional Euclidean space there exists a 28-dimensional algebra of linear vector elds tangent to concentric 7-spheres, and this is the Lie algebra (by which the corresponding Lie group is reconstructed) of the classical Cliord algebra Cl0,3 (R). It is possible that the elements of this algebra are representatives of the simplest and most symmetric form of matter motion, which is realized in nature only at the beginning (end) of its evolutionary development. On the other hand, since at the moment of evolution the static vacuum vector eld is tangential to the product of S 3 × S 3 spheres, it can be assumed that a 3-dimensional sphere of larger diameter forms a 3-dimensional Euclidean space of the observer, and a 3-dimensional sphere of smaller diameter forms internal symmetries of elementary particles. Indeed, the linear vector elds of the 4-dimensional Euclidean space tangent to the 3dimensional spheres of this space form the Lie algebra of the quaternion algebra into which the 3-dimensional Euclidean space is embedded. At the same time, since the topological features of the vector eld, having the shape of a circle (ring), move along the surface of the toroidal variety, it makes sense to pay attention to the symmetry of the torus entwined around the sphere with the punched poles. In this regard, we note that any special unitary group
9
SU (n) is generated by the products of matrices of the form OT O0 , where the matrices O, O0 belong to the rotation group n - dimensional sphere SO(n, R), and the matrix T belongs to the group of ring motions on the torus SD(n, C), which consists of special (in the sense of equality to the determinant unit) diagonal matrices with elements of the form eiϕj . Throughout earlier, simplifying the essence of the question, we assumed that the topological feature of a vector eld is a circle (ring). However, in fact, the current lines of a closed vector eld of a toroidal variety corresponding to the zero vector eld of an 8-dimensional space with a neutral metric are only in the simplest case closed in a circle, and the question of classi cation of topological features and their correspondence to elementary particles remains open. Thus, we have an answer to the question of where matter moves from the point of view of an external observer. It remains to answer the question of how it moves from the point of view of an external observer. Let there be a unit sphere S 7 in the 8-dimensional Euclidean space and a vector eld u(x) such that (u(x), x) = 0. Then we can easily get the functionality Z (0.32) J = u2 (x)d7 s, S7
where u(x) is an arbitrary tangent to S 7 vector velocity eld of matter particles whose zero variation limits the arbitrariness u(x). In this case, when the vector eld u(x) is potential, i.e. u(x) = ∇ϕ(x), our variational problem has a necessary condition in the form of a system of dierential equations: ( ∆ϕ(x) = 0 (0.33) ∇x ϕ(x) = 0,
= 0 is reduced which in spherical coordinates by virtue of the fact that ∂ϕ dρ to the Laplace equation on the sphere, and hence ϕ(x) is a constant, since, as is known [?], [?], the only harmonic function on the sphere is constant. However, if the evolutionary change of the vector eld ∇ϕ(x) given on the sphere is represented by the vector eld of the sphere, then the radius of the sphere r, according to the accepted idealization, increases during the evolution of the vector eld ∇ϕ(x) together with the evolutionary parameter τ , and the second equation of the system should be excluded, replacing it 10
with an algebraic inequality
( ∆ϕ(x) = 0 x2 ≤ r.
(0.34)
And since in the time interval τ and τ + ∆τ the boundary conditions of the problem lie on the sphere of radius rτ and rτ +∆τ , then as the simplest solutions of the Laplace equation in the ball layer of the eight-dimensional Euclidean space we should use ball harmonic functions, which are divided into radial and spherical parts. Thus, spherical functions and their arbitrary linear combinations given on the sphere S 7 can be interpreted as potential functions of the static (independent of the evolutionary parameter) vector eld of moving matter. However, the direction of evolution of the vector eld uτ (x) could be given by an inequality Z Z
u2τ (x)d7 s >
S7
u2τ +∆τ (x)d7 s,
(0.35)
S7
and the uctuations of the ow of moving matter are interpreted as deviations from the variational principle that minimizes the functional Z τ J = u2τ (x)d7 s. (0.36) S7
Moreover, since the lamination of globally minimal (vacuum) ow of moving matter is expected to have a topology of R3 × S 3 variety, the uctuations can be localized in the Euclidean space R3 as local violations of the variational principle. So, we answered the main questions, and thus outlined the direction of future studies of moving matter the study of dynamic ows on the sphere S 7 . However, all the further (main) content of this monograph is a step back, which only prepares the mathematical ground for the achievement of the highest goal, and for the formal presentation of the content of the book should refer to the following list of annotations:
• In the article 'Some applications of algebra of vector elds' discusses the local algebra of linear vector elds, which are used in the mathematical modeling of the physical space of the dynamic ows of vector elds on 8-dimensional space with a neutral metric that is wrapped by the 11
factorization of the isotropic cone. It is shown that topological features of vector elds obey Dirac equations.
• In the paper 'On group constructions on the product of spheres' we investigate the topological connection between Abelian and nabalebale groups of parity. Abelian groups of parity are formed as kernels of homomorphisms of parity in group Zn and non-Abelian parity groups are formed as kernels of homomorphisms of parity in group S2 oSn . Node factorization the integer lattice using the Abelian parity group we we obtain a quotient lattice that serves as a one-dimensional cell complex (framework) of the corresponding product of spheres. It is shown that the automorphisms of this quotient lattice form a corresponding nonAbelian parity group. • In the article 'On parallelepipeds in algebra and topology' we assume that the concept of parallelepipeds application can be found in such branches of mathematics as combinatorial analysis, multilinear algebra and algebraic topology. Indeed, the edge sequence n-dimensional parallelepiped spanned by the basis n-dimensional space, isomorphic to the permutations. In turn, the calculation of the oriented volume of the parallelepiped as a polyline function that takes a zero value on linearly dependent vectors leads to the concept of a alternating polyline product. Finally, parallelepipeds can easily fold into cell spaces and form chain complexes with a homology group as a topological invariant of these spaces. Having in mind the above-mentioned penetrations of geometry into algebra and topology, we will consistently develop an algebra-topological formalism related to parallelepipeds, and then nd its application in solving the problem of classi cation of closed orientable varieties of arbitrary dimension. • The article 'On the construction of vector elds of spheres' shows how to form the maximum possible system of linearly independent linear vector elds of an odd-dimensional sphere of any dimension with the help of a group of mirror symmetries. • In the article 'On sphere winding' we rst bring the reader to one remarkable result of the action of the modular group on the sphere, proving that the modular group allocates on the winding sphere, the longitude of which is a function on the set of Prime numbers, and then, 12
considering the dynamics of windings, we notice that in the problem of random walk along the broken lines of the winding of the sphere, the concept of complex probability amplitude arises quite naturally, and the dynamics of the probability of the wandering particle obeys the generalized schrodinger equation. It remains only to add that this small volume brochure is addressed not only to readers who are able to feel the philosophical and mathematical idea of the author, but also skeptical readers who may be interested only in the mathematical aspects of this idea.
13
