Incomplete Delta Functions

By applying projection operators to state vectors of coordinates we obtain subspaces in which these states are no longer normalized according to Dirac's delta function but normalized according to what we call "incomplete delta functions". We show that this class of functions satisfy identities similar to those satisfied by the Dirac delta function. The incomplete delta functions may be employed advantageously in projected subspaces and in the link between functions defined on the whole space and the projected subspace. We apply a similar procedure to finite dimensional vector spaces for which we define incomplete Kronecker deltas. Dispersion relations for the momenta are obtained and ''sums over poles'' are defined and obtained with the aid of differences of incomplete delta functions.Comment: 11 pages, LaTe

The RR interval spectrum, the ECG signal and aliasing

A reliable spectral analysis requires sampling rate at least twice as large as the frequency bound, otherwise the analysis will be unreliable and plagued with aliasing distortions. The RR samplings do not satisfy the above requirements and therefore their spectral analysis might be unreliable. In order to demonstrate the feasibility of aliasing in RR spectral analysis, we have done an experiment which have shown clearly how the aliasing was developed. In the experiments, one of us (A.G) had kept his high breathing rate constant with the aid of metronome for more than 5 minutes. The breathing rate was larger than one-half the heart rate. Very accurate results were obtained and the resulting aliasing well understood. To our best knowledge this is the first controlled experiment of this kind coducted on humans. We compared the RR spectral analysis with the spectrum of the ECG signals from which the RR intervals were extracted. In the significant for RR analysis frequencies (below one-half Hertz) significant differences were observed. In conclusion we recommend to study the spectral analysis of the ECG signal in the free of aliasing frequency range.Comment: 27 pages, 12 figure

Topology of the Standard Model, I: Fermions

The Harari-Shupe model for fermions is extended to a topological model which contains an explanation for the observed fact that there are only three generations of fermions. Topological explanations are given for $\beta$-decay and for proton decay predicted in supersymmetry and string theories. An explanation is given for the observed fact that the three generations of fermions have such similar properties. The concept of "color" is incorporated into the model in a topologically meaningful way. Conservation laws are defined and discussed in the context of the algebraic topology of the model, and preon number is proved to be linearly determined by charge, weak isospin, and color.Comment: 39 pages, 9 figures. A "disclaimer" has been added at the request of the editors of the journal to which the article has been submitted for publication. The new version of 12/26/2015 adds new appendices. One contains a more intuitive setting meant for those physicists who objected to the van Kampen diagrams of the original and another contains a geometric model for quantum entanglemen

Tensor Lagrangians, Lagrangians equivalent to the Hamilton-Jacobi equation and relativistic dynamics

We deal with Lagrangians which are not the standard scalar ones. We present a short review of tensor Lagrangians, which generate massless free fields and the Dirac field, as well as vector and pseudovector Lagrangians for the electric and magnetic fields of Maxwell's equations with sources. We introduce and analyse Lagrangians which are equivalent to the Hamilton-Jacobi equation and recast them to relativistic equations.Comment: 12 page

Maxwell equations as the one-photon quantum equation

Maxwell equations (Faraday and Ampere-Maxwell laws) can be presented as a three component equation in a way similar to the two component neutrino equation. However, in this case, the electric and magnetic Gauss's laws can not be derived from first principles. We have shown how all Maxwell equations can be derived simultaneously from first principles, similar to those which have been used to derive the Dirac relativistic electron equation. We have also shown that equations for massless particles, derived by Dirac in 1936, lead to the same result. The complex wave function, being a linear combination of the electric and magnetic fields, is a locally measurable and well understood quantity. Therefore Maxwell equations should be used as a guideline for proper interpretations of quantum theories.Comment: 9 pages, LaTe

Asphericity for certain groups of cohomological dimension 2

A finite connected 2-complex K whose fundamental group is of cohomological dimension 2 is aspherical iff the subgroup \Sigma_K of H_2(K) consisting of spherical 2-cycles is zero. A finite connected subcomplex of an aspherical 2-complex is aspherical iff its fundamental group is of cohomological dimension 2. If G is a countable group such that extension of scalars from Z[G] to \ell_2(G) kills \bar K_0(Z[G]), and if P is a finitely generated projective Z[G]-module with P/IP=0, where I is the augmentation ideal of Z[G], then P=0. In particular, if G is a countable group of cohomological dimension 2 and P is a finitely generated projective Z[G]-module such that P/IP=0, then P=0.Comment: 11 pages. The new version incorporates results due to B. Eckmann, fixes an omission, and corrects some typo

Orthogonality and Boundary Conditions in Quantum Mechanics

One-dimensional particle states are constructed according to orthogonality conditions, without requiring boundary conditions. Free particle states are constructed using Dirac's delta function orthogonality conditions. The states (doublets) depend on two quantum numbers: energy and parity. With the aid of projection operators the particles are confined to a constrained region, in a way similar to the action of an infinite well potential. From the resulting overcomplete basis only the mutually orthogonal states are selected. Four solutions are found, corresponding to different non-commuting Hamiltonians. Their energy eigenstates are labeled with the main quantum number n and parity "+" or "-". The energy eigenvalues are functions of n only. The four cases correspond to different boundary conditions: (I) the wave function vanishes on the boundary, (II) the derivative of the wavefunction vanishes on the boundary,(III) periodic (symmetric) boundary conditions, (IV) periodic (antisymmetric)boundary conditions . Among the four cases, only solution (III) forms a complete basis in the sense that any function in the constrained region, can be expanded with it. By extending the boundaries of the constrained region to infinity, only solution (III) converges uniformly to the free particle states. Orthogonality seems to be a more basic requirement than boundary conditions. By using projection operators, confinement of the particle to a definite region can be achieved in a conceptually simple and unambiguous way, and physical operators can be written so that they act only in the confined region.Comment: 10 pages, LaTe

Peculiarities of Brain's Blood Flow : Role of Carbon Dioxide

Among the major factors controlling the cerebral blood flow (CBF), the effect of PaCO2 is peculiar in that it violates autoregulatory CBF mechanisms and allows to explore the full range of the CBF. This research resulted in a simple physical model, with a four parameter formula, relating the CBF to PaCO2. The parameters can be extracted in an easy manner, directly from the experimental data. With this model earlier experimental data sets of Rhesus monkeys and rats were well fitted. Human data were also fitted with this model. Exact formulae were found, which can be used to transform the fits of one animal to the fits of another one. The merit of this transformation is that it enable us the use of rats data as monkeys data simply by rescaling the PaCO2 values and the CBF data. This transformation makes possible the use of experimental animal data instead of human ones.Comment: 24 pages, 5 figure

Filling Length in Finitely Presentable Groups

Filling length measures the length of the contracting closed loops in a null-homotopy. The filling length function of Gromov for a finitely presented group measures the filling length as a function of length of edge-loops in the Cayley 2-complex. We give a bound on the filling length function in terms of the log of an isoperimetric function multiplied by a (simultaneously realisable) isodiametric function.Comment: 10 pages, 3 figure

Consistent quantization of massless fields of any spin and the generalized Maxwell's equations

A simplified formalism of first quantized massless fields of any spin is presented. The angular momentum basis for particles of zero mass and finite spin s of the D^(s-1/2,1/2) representation of the Lorentz group is used to describe the wavefunctions. The advantage of the formalism is that by equating to zero the s-1 components of the wave functions, the 2s-1 subsidiary conditions (needed to eliminate the non-forward and non-backward helicities) are automatically satisfied. Probability currents and Lagrangians are derived allowing a first quantized formalism. A simple procedure is derived for connecting the wave functions with potentials and gauge conditions. The spin 1 case is of particular interest and is described with the D^(1/2,1/2) vector representation of the well known self-dual representation of the Maxwell's equations. This representation allows us to generalize Maxwell's equations by adding the E_0 and B_0 components to the electric and magnetic four-vectors. Restrictions on their existence are discussed.Comment: IARD 201