arXiv subjects
Publications and source records attributed to Gordon Chalmers.
The inversion of an FIR data sampling is usually stated to be possible with the use of a potentially unstable IIR, and in particular circumstances. It is possible to accomplish the same inversion with the doubling of an FIR sampling and with only FIRs for the sampling and the inversion. This note presents the configuration, which appently is not in the literature, for perfect signal reconstruction.
Amplitudes $A_n$ in $d$-dimensional scalar field theory are generated, to all orders in the coupling constant and at $n$-point. The amplitudes are expressed as a series in the mass $m$ and coupling $λ$. The inputs are the classical scattering, and these generate, after the integrals are performed, the series expansion in the couplings $λ_i$. The group theory of the scalar field theory leads to an additional permutation on the $L$ loop trace structures. Any scalar field theory, including those with higher dimension operators and in any dimension, are amenable.
Metrics on Calabi-Yau manifolds are used to derive a formula that finds the existence of integer solutions to polynomials. These metrics are derived from an associated algebraic curve, together with its anti-holomorphic counterpart. The integer points in the curve coincide with points on the manifold, and the metric form around these points are used to find their existence. The explicit form of the metrics can be found through a solution to the D-terms in a non-linear sigma model.
All of the known fermions fit a simple formula which is presented here. The simple formula appears to indicate a topological origin to mass generation, and it is globally accurate to an approximate percent. Instantons or a quantized two Higgs mechanism, as in the MSSM model, could give a simple origin to the masses.
The quantum effective action may be used to invert information from phenomena, either measured or ideal, to the microscopic Lagrangian. As an example of this procedure the lattice composition of a solid can be determined in principle from desired critical phenomena. Also, the engineering of scattering and optical properties of particular degrees of freedom, and excitations, in terms of the temperature and electro-magnetic fields may be performed.
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
N=2 gauged non-linear sigma models are examined classically and their D-terms are solved. The variation of the classical Lagrangian in order to solve for the auxiliary fields is identical to integrating these modes functionally. The latter is performed for the general quotient. The D-term solution is equivalent to solving a coupled set of algebraic equations.
A generating function is derived that counts the number of diagrams in an arbitrary scalar field theory. The number of graphs containing any number $n_j$ of $j$-point vertices is given. The count is also used to obtain the number of classical graphs in gauge theory and gravity.
The derivative expnsion in the context of IIB string scattering compactified on non-trivial K3 and other Calabi-Yau manifolds is formulated. The scattering data in terms of automorphic functions can be inverted to find the these metrics. The solutions are parameterized by the moduli information, and the metrics may be found to any desired accuracy in derivatives. Metric information to low orders in derivatives allows for a counting of curves inside the manifold; in addition, the coefficients of these exponential terms via D-brane wrappings are polynomials that may admit an invariant interpretation in cohomology. An interesting case pertaining to M-theory compactifications is the collection of seven-dimensional $G_2$ manifolds; they can also be obtained when the moduli space degenerates into cases, such as a toroidal one or other limit in which modular functions on the space are known. Note: this work was written two years ago; the recipe without the explicit form of the scattering and metrics is given.
A characterization of instanton contributions to gauge field theory is given. The dynamics of instantons with the gluon field is given in terms of 'classical' instanton contributions, i.e. on the same footing as tree amplitudes in field theory. This paramaterization depends on the kinematics of the local instanton operators and their coupling dependence. A primary advantage of this 'classical' formulation is that the weak-strong duality and its relations with the perturbative sector are possibly made more manifest.
The terms in the quantum scattering in scalar field theory models is parameterized by the invariants $\prod s_{ij}^{n_{ij}}$. The $s_{ij}$ are kinematic two-particle invariants, and the $n_{ij}$ are integers. The coefficients of these terms are computed via integrating all Feynman diagrams, or within the derivative expansion by solving the iteration equations. The latter has been provided recently; the functions which are prefactors of the individual terms $\prod s_{ij}^{n_{ij}}$ can be interpreted as terms in the expansions of L-series, which may be specified by collections of their zeroes. Once finding the appropriate elliptic curve coefficients, these quantum field solutions provide an algorithm to determining all of the mod p zeros to the algebraic curves. The latter is presumably determined by 'experimental' computer modeling or by the appropriate determination of the quantum prefactors.
The $n$-point amplitudes of gauge and gravity theory are given as a series in the coupling. The recursive derivative expansion is used to find all of the coupling coefficients. Initial conditions to any bare Lagrangian, or of an improved action, are used to compute quantum amplitudes.
Gauge theory amplitudes in a non-helicity format are generated at all $n$-point and at tree level. These amplitudes inherit structure from $ϕ^3$ classical scattering, and the string inspired formalism is used to find the tensor algebra. All of the classical gravity amplitudes are also given. The classical effective action can also be constructed. Generalizations to amplitudes with non spin-1 or 2 is possible.
The polytopic definition introduced recently describing the topology of manifolds is used to formulate a generating function pertinent to its topological properties. In particular, a polynomial in terms of one variable and a tori underlying this polynomial may be defined that generates an individual cohomological count. This includes the de Rham complex for example, as well as various index theorems by definition such as homotopy. The degree of the polynomials depends on the volume used to define the region parameterizing the manifolds; its potentially complex form and L-series is not presented in this work. However, the polynomials and the relevant torii uniformize the topological properties in various dimensions; in various dimensions this is interesting in view of known topologies.
A scheme is presented based on numbers that represent a manifold in $d$ dimensions for generalizations of textbook cryptosystems. The interlocking or intersection of geometries, requiring the addition of a series of integers $q_j$, can be used to enhance a cryptosystem via algorithms based on the form of the geometry. Further operations besides addition of these numbers, e.g. sewing of the geometries, can be given such as rotations or contractions.
The tree amplitudes in scalar field theories are presented at all $n$. The momentum routing of propagators is given at $n$-point in terms of a specified set of numbers, and the mass expansion of the massive theories is generated. A group structure on the diagrams is given. The tree amplitudes can be used to find the effective action.
A construction of polytopes is given based on integers. These geometries are constructed through a mapping to pure numbers and have multiple applications, including statistical mechanics and computer science. The number form is useful in topology and has a mapping to one-dimensional knot contours.
A route to the derivation of the numbers $s$ to the transcendental equation $ζ(s)=0$ is presented. The solutions to this equation require the solving of a geodesic flow in an infinite dimensional manifold. These solutions enable one approach to a formula generating the prime numbers.