I think one of the things that helps scientific revolution is when the places it's going into is desolate. So you know it helps when things are kind of burnt out and there's this new spark of life that can expand into a vacuum instead of having to push back existing ideas in that area. So that would be my impression about what's involved there. You get this effective acceleration, photon propagation through variable-dimensional space and "how to make the world a better place" ...the fact is that like other systems that have computational irreducibility when you set down a system with rules the system has a computationally irreducible behavior that it will follow that will always lead to unexpected things happening. My perfect scheme will always have a bug. The intensity of the EM wave is either amplified or damped, so when you see us under the stars, our recent continuum results are a little less brittle. They seem to work better than approaches that are more constrained and we seem to know what's "going" to happen.
simpleGridGraph =
GridGraph[{32, 32}, VertexSize -> Small, ImageSize -> Large];
GetAmplitudes[g_] := AnnotationValue[g, VertexWeight];
Attributes[SetAmplitudes] = {HoldFirst};
SetAmplitudes[g_, amplitudes_] :=
AnnotationValue[g, VertexWeight] = amplitudes;
SetDeltaFunction[g_, vertex_, distance_] :=
Module[{ng =
g}, (AnnotationValue[{ng, #}, VertexWeight] =
ConstantArray[1, Length[#]]) &@
VertexList[NeighborhoodGraph[g, vertex, distance]];
ng];
UnnormalizedLaplacian[g_] :=
N@(DiagonalMatrix[VertexOutDegree[#]] - AdjacencyMatrix[#]) &@
UndirectedGraph[g];
GraphSpectrum[
g_] := <|"eigenvalues" -> Eigenvalues[#],
"eigenvectors" -> Eigenvectors[#]|> &@UnnormalizedLaplacian[g];
EigenSpaceAdvance[spectrum_, coordinates_,
t_] := (Cos[1/10*t*spectrum["eigenvalues"]] +
Sin[1/10*t*spectrum["eigenvalues"]])*coordinates;
AmplitudeSpaceAdvance[spectrum_, ic_, t_] :=
Transpose[spectrum["eigenvectors"]] .
EigenSpaceAdvance[spectrum, spectrum["eigenvectors"] . ic, t];
Attributes[TimePropagate] = {HoldFirst};
TimePropagate[g_, t_] :=
SetAmplitudes[g,
AmplitudeSpaceAdvance[GraphSpectrum[g], GetAmplitudes[g], t]];
SetAmplitudes[simpleGridGraph, ConstantArray[0, 1024]];
deltaGridGraph =
SetDeltaFunction[simpleGridGraph,
First[GraphCenter[simpleGridGraph]], 1];
AmplitudeMatrix2D[g_, n_] := ArrayReshape[GetAmplitudes[g], {n, n}];
SnowconeColor[z_] :=
Blend[{RGBColor[0, 0, 1], RGBColor[1, 1, 1], RGBColor[1, 0.4, 0.7]},
z];
animation3DTable = Table[TimePropagate[deltaGridGraph, t];
ListPlot3D[AmplitudeMatrix2D[deltaGridGraph, 32],
ColorFunction -> SnowconeColor,
PlotRange -> {{1, 32}, {1, 32}, All},
BoxRatios -> {1, 1, 0.5}], {t, 0, 10, 0.25}];
ListAnimate[animation3DTable]
The Simon Fischer method to model the propagation of a classical photon represented as a transversal electromagnetic (EM) wave...through a space that changes in dimensionality and is discretized on a graph...Plato had approaches to, this is how to set up the perfect system of government. The king of philosophy, what aristocracy may particularly..the ability of history and by necessity..any aspects of society successfully have a habit of repeating themselves..one country will puff out and pop and it's like what you could imagine. The computational irreducibility bites you, what with this and that thorough exploration that we have made, of the wave equation on graphs, whether it's the use of the wave equation and I use Mathematica all the time to go do exploration of the wave equation on graphs, particularly the use of the Laplacian Matrix and the sound of discretizing continuum solutions. It's the look that we have into binary code, heterodimensional graphs, which are chains of different dimensions connected via methods like corner docking or face docking. It's been known that the concept of fractional dimensions is addressed with intrinsic fractional graphs (like the SierpiÅ„ski graph) and extrinsic methods by..labeling certain regions to indicate a dimension change.
SierGraph[n_, d_] :=
MeshConnectivityGraph[SierpinskiMesh[n, d], 0,
VertexLabels -> Automatic];
sierpinskiGraph = SierGraph[2, 3];
PhotonPropagation[g_, steps_] :=
Module[{amplitudes = ConstantArray[0, VertexCount[g]], photonPos},
photonPos = RandomChoice[VertexList[g]];
amplitudes[[photonPos]] = 1;
Table[photonPos = RandomChoice[AdjacencyList[g, photonPos]];
amplitudes[[photonPos]] = 1;
HighlightGraph[g, photonPos, VertexSize -> Large], {steps}]];
SimulateEffects[g_, effectType_, steps_] :=
Module[{propagationFunction},
propagationFunction =
Switch[effectType, "DarkMatter", PhotonPropagation, "DarkEnergy",
PhotonPropagation, "BlackHoles", PhotonPropagation,
"BosonicStars", PhotonPropagation, "NeutrinoTravel",
PhotonPropagation, _, PhotonPropagation];
ListAnimate[propagationFunction[g, steps]]];
animationPhoton = ListAnimate[PhotonPropagation[sierpinskiGraph, 30]]
animationDarkMatter =
SimulateEffects[sierpinskiGraph, "DarkMatter", 30]
animationOfDarkEnergy =
SimulateEffects[sierpinskiGraph, "DarkEnergy", 30]
animateBlackHoles =
SimulateEffects[sierpinskiGraph, "BlackHoles", 30]
animatedNeutrinoTravel =
SimulateEffects[sierpinskiGraph, "NeutrinoTravel", 30]
animationBosonicStars =
SimulateEffects[sierpinskiGraph, "BosonicStars", 30]
If you set up the binary numbers in a fixed number of digits, you have this tree structure where you say it's a 1-branch or a 0-branch and that, is what happens with the hexagons, the concept of fractional dimensions and curvature that I discuss as a model for fractional dimensions, with the introduction of curvature into graphs mimicking dimension change. Because presumably if you do an infinite number of batches you'll get, eventually, one particle that just by "luck", just perfectly..just goes the exact way. To generalize the Laplacian Matrix to fractional dimensions, alone is going to need to require none of the fractional dimensions implicitly..but first I kind of investigate the role of curvature in modeling non-integer dimensions. Let's say the users "deplete" the simulation of photon propagation into various hypothetical scenarios, including travel through dark matter, dark energy, near black holes, bosonic stars, and neutrino travel. All of those computational marbles that I have dropped case in point, brings about a generous way of getting these hypothetical scenarios whether it's setting up an animating these scenarios, fully capturing the dynamics of wave propagation in fractional dimensions..and the idea of Lyapunov exponents in studying control systems had this idea that you can measure the exponential divergence, and when things are stable the thing will damp out our exponent, exponential divergence is like excavating lower and lower order digits, smaller and smaller digits in the number so to speak as time goes on. So in the 1930s, in the origin of this thing called symbolic dynamics, specifically Garrett Birkhoff..mainstream mathematics where Marson Morse, Konrad Zuse, had a clear choice. We've already developed a mathematical framework that extends the definition of the Laplacian in non-integer dimensions and studying its spectral properties. This is self-explanatory, how curvature introduced to graphs can prettify and simulate fractional dimensions and lead to new insights in geometric analysis and graph theory. And apparently we can simulate something like physical phenomena, just as a shortcut for accessing the black hole event horizons garbage "dump" and bosonic stars which involve incorporating general relativity and quantum field theory into the graph-based framework, pictures of octopuses and so on and base 8..so at that time, it certainly was a slightly obscure piece of knowledge, something that was part of the engineering of computers but not part of the everyday, ordinary, world like ours.
Attributes[SetAmplitudes] = {HoldFirst};
Attributes[TimePropagate] = {HoldFirst};
simpleGridGraph = GridGraph[{32, 32}, VertexSize -> Small];
GetAmplitudes[g_] := AnnotationValue[g, VertexWeight];
SetAmplitudes[g_, amplitudes_] :=
AnnotationValue[g, VertexWeight] = amplitudes;
SetDeltaFunction[g_, vertex_, distance_] :=
Module[{ng =
g}, (AnnotationValue[{ng, #}, VertexWeight] =
ConstantArray[1, Length[#]]) &@
VertexList[NeighborhoodGraph[g, vertex, distance]];
ng];
UnnormalizedLaplacian[g_] :=
N@(DiagonalMatrix[VertexOutDegree[g]] - AdjacencyMatrix[g]) &@
UndirectedGraph[g];
GraphSpectrum[
g_] := <|"eigenvalues" -> Eigenvalues[#],
"eigenvectors" -> Eigenvectors[#]|> &@UnnormalizedLaplacian[g];
EigenSpaceAdvance[spectrum_, coordinates_,
t_] := (Cos[1/10*t*spectrum["eigenvalues"]] +
Sin[1/10*t*spectrum["eigenvalues"]])*coordinates;
AmplitudeSpaceAdvance[spectrum_, ic_, t_] :=
Transpose[spectrum["eigenvectors"]] .
EigenSpaceAdvance[spectrum, spectrum["eigenvectors"] . ic, t];
TimePropagate[g_, t_] :=
SetAmplitudes[g,
AmplitudeSpaceAdvance[GraphSpectrum[g], GetAmplitudes[g], t]];
SetAmplitudes[simpleGridGraph, ConstantArray[0, 1024]];
deltaGridGraph =
SetDeltaFunction[simpleGridGraph,
First[GraphCenter[simpleGridGraph]], 1];
AmplitudeMatrix2D[g_, n_] := ArrayReshape[GetAmplitudes[g], {n, n}];
SnowconeColor[z_] :=
Blend[{RGBColor[0, 0, 1], RGBColor[1, 1, 1], RGBColor[1, 0.4, 0.7]},
z];
animation3DTable = Table[TimePropagate[deltaGridGraph, t];
ListPlot3D[AmplitudeMatrix2D[deltaGridGraph, 32],
ColorFunction -> SnowconeColor,
PlotRange -> {{1, 32}, {1, 32}, {-0.1, 1.2}},
BoxRatios -> {1, 1, 0.5}], {t, 0, 5, 0.1}];
ListAnimate[animation3DTable]
So there's this implicit regularization inherent to Fischer's methodology, the architecture that serves as the foundation of intelligent modeling of the influence of dark matter and dark energy on photon propagation, with an elegant construction of graph models that is going to cut it - the gravitational effects predicted by theories of dark matter and energy distributions in the universe. When he said "bit" I thought it was sort of a binary choice that leads to everything, that was actually the idea of subsequent efforts this choice of distinction that makes everything. So in the midst of that, the graph approach could be adapted to simulate neutrino oscillations and their propagation through space, that's the implicit perspective on particle physics because "no matter how" much "corruption" is inherent in the data, the hypergraphs within the Wolfram Model of physics uses hypergraphs to model the evolution of space-time. What really matters is the interaction between fractional dimensionality and hypergraph rewriting rules, which could be a fruitful area of research in the randomness or unstructuredness or the meaningfulness of the algorithmic data which "historically" simulates photon propagation in variable-dimensional spaces, supporting the simulation of the Wolfram Language.
Attributes[SetAmplitudes] = {HoldFirst};
Attributes[TimePropagate] = {HoldFirst};
simpleGridGraph = GridGraph[{32, 32}];
GetAmplitudes[g_] := AnnotationValue[g, VertexWeight];
SetAmplitudes[g_, amplitudes_] :=
AnnotationValue[g, VertexWeight] = amplitudes;
SetDeltaFunction[g_, vertex_, distance_] := Module[{ng = g},
(AnnotationValue[{ng, #}, VertexWeight] =
ConstantArray[1, Length[#]]) &@
VertexList[NeighborhoodGraph[g, vertex, distance]];
ng];
UnnormalizedLaplacian[g_] :=
N@(DiagonalMatrix[VertexOutDegree[g]] - AdjacencyMatrix[g]) &@
UndirectedGraph[g];
GraphSpectrum[g_] := <|"eigenvalues" -> Eigenvalues[#],
"eigenvectors" -> Eigenvectors[#]|> &@
UnnormalizedLaplacian[g];
EigenSpaceAdvance[spectrum_, coordinates_, t_] := (
Cos[1/10*t*spectrum["eigenvalues"]] +
Sin[1/10*t*spectrum["eigenvalues"]]
)*coordinates;
AmplitudeSpaceAdvance[spectrum_, ic_, t_] :=
Transpose[spectrum["eigenvectors"]] .
EigenSpaceAdvance[spectrum, spectrum["eigenvectors"] . ic, t];
TimePropagate[g_, t_] := SetAmplitudes[g,
AmplitudeSpaceAdvance[GraphSpectrum[g], GetAmplitudes[g], t]];
SetAmplitudes[simpleGridGraph, ConstantArray[0, 1024]];
deltaGridGraph = SetDeltaFunction[simpleGridGraph,
First[GraphCenter[simpleGridGraph]],
1];
AmplitudeMatrix2D[g_, n_] := ArrayReshape[
GetAmplitudes[g],
{n, n}];
SnowconeColor[z_] := Blend[{
RGBColor[0, 0, 1],
RGBColor[1, 1, 1],
RGBColor[1, 0.4, 0.7]}, z];
animation3DTable = Table[TimePropagate[deltaGridGraph, t];
ListPlot3D[
AmplitudeMatrix2D[deltaGridGraph, 32],
ColorFunction -> SnowconeColor,
PlotRange -> {
{1, 32}, {1, 32},
{-10000, 10000}
}, BoxRatios -> {1, 1, 0.5}],
{t, 0, 10, 0.25}];
la = ListAnimate[animation3DTable]
@Simon Fischer When you see the propagation of the transversal EM wave through variable-dimensional space and yeah. I don't know, but you can see it in the intensity of the wave that gets amplified or damped depending on the dimensional change..the recent results in the continuum. Okay, so I love all this interpretation of interactive educational modules that we have made as a springboard to help students and enthusiasts learn about complex concepts in physics and mathematics. So all this "interpretation" is just a few simple mathematical equations, such as biological networks or social structures and how they can provide new methods for analyzing "agnostic" system behaviors. Now "quickly" exploring your variable-dimensional graph models, in computational geometry...has allowed me to descend gradient-wise through theoretical physics and beyond. Applied mathematics and computational science and its connections to the Observer and the Ruliad. The primary cybernetics, focus of the research on controllability so that they stay within a certain required boundary, because if you place constraints in the right way you can decay dark matter particles that result in the emission of photons, which would be indicated by a narrow line-like spectral feature. Of 32 line-like features detected above a 3sigma significance, 29 coincide with known instrumental and astrophysical lines. The ON-OFF approach which constrains the lifetime of dark matter particles with masses ranging from 4 keV to 14 MeV is greater than 10^20 - 10^21 years. How can that be right? Oh, boy the updated constraints that you provide on dark matter decay, iteratively refine our understanding of the properties of dark matter particles. Systematic uncertainties fluctuate and bounce down your gradient descent because your gradient descent triggers a lot of scientific interpretation in a rather different direction. It's sort of like those two decades of data from the INTEGRAL spacecraft's SPI spectrometer. Lots of people getting together to talk about this problem of how all these systems like animals get to do, what they do. And you can load this popup, no YouTube just the mixing angle of sterile neutrino dark matter that is (and/or) its limits when all that stuff in Wolfram Language runs off the screen. So we've got that.
simpleGridGraph = GridGraph[{32, 32}];
GetAmplitudes[g_] := AnnotationValue[g, VertexWeight];
Attributes[SetAmplitudes] = {HoldFirst};
SetAmplitudes[g_, amplitudes_] :=
AnnotationValue[g, VertexWeight] = amplitudes;
SetDeltaFunction[g_, vertex_, distance_] := Module[{ng = g},
(AnnotationValue[{ng, #}, VertexWeight] =
ConstantArray[1, Length[#]]) &@
VertexList[NeighborhoodGraph[g, vertex, distance]];
ng];
UnnormalizedLaplacian[g_] :=
N@(DiagonalMatrix[VertexOutDegree[#]] - AdjacencyMatrix[#]) &@
UndirectedGraph[g];
GraphSpectrum[g_] := <|"eigenvalues" -> Eigenvalues[#],
"eigenvectors" -> Eigenvectors[#]|> &@
UnnormalizedLaplacian[g];
EigenSpaceAdvance[spectrum_, coordinates_,
t_] := (Cos[2*t*spectrum["eigenvalues"]])*coordinates;
AmplitudeSpaceAdvance[spectrum_, ic_, t_] :=
Transpose[spectrum["eigenvectors"]] .
EigenSpaceAdvance[spectrum,
spectrum["eigenvectors"] . ic,
t];
Attributes[TimePropagate] = {HoldFirst};
TimePropagate[g_, t_] := SetAmplitudes[g,
AmplitudeSpaceAdvance[GraphSpectrum[g],
GetAmplitudes[g],
t]];
SetAmplitudes[simpleGridGraph, ConstantArray[0, 1024]];
deltaGridGraph = SetDeltaFunction[
simpleGridGraph,
First[GraphCenter[simpleGridGraph]],
1];
AmplitudeMatrix2D[g_, n_] := ArrayReshape[GetAmplitudes[g], {n, n}];
SnowconeColor[z_] := Blend[{
RGBColor[0, 0, 1],
RGBColor[1, 1, 1],
RGBColor[1, 0.4, 0.7]}, z];
animation3DTable = Table[TimePropagate[deltaGridGraph, t];
ListPlot3D[
AmplitudeMatrix2D[deltaGridGraph, 32],
ColorFunction -> SnowconeColor,
PlotRange -> {{1, 32}, {1, 32}, {-0.1, 1.2}},
BoxRatios -> {1, 1, 0.5}],
{t, 0, 5, 0.1}];
Export["animation3DTable2.gif", animation3DTable]
ListAnimate[animation3DTable]
People were talking about Cybernetics and Cyber everything..the Doctor Who program, might be classified as science fiction might be classified as comedy but it's changed over the course of time this "must have been" 1965.. what I call it, my broken tape recorder. That is when, we will computerize everything. I will show you, how complex behavior arises from simple rules: a sterile neutrino decays into a photon and a neutrino. And it has "no mass" so you can do, whatever you want, to do. Look at me. The best functions were the Laplacian
and the ColorByInternalamplitude
.
The observation of dark matter decay signals within the SPI data requires a complex understanding of both the observer's capabilities and limitations. Your constraints on decaying dark matter provide the "architectural" foundation of computational analysis of data, and Wolfram's emphasis on computation as a powerful tool to formalize the world. Spectral lines that is their analysis is implicitly a multiway system, where each line represents a potential path of history for a decaying dark matter particle. In the context of quantum mechanics and modeling growth processes, akin to the computational processes Wolfram describes, we can find this whole "American induction" technique to use generative AI to simulate the perception of alien minds where the attempt is to understand and model dark matter decay phenomena that are not directly observable, much like an alien process to our everyday, practical, experience of what we are doing "right now". The progression of Mathematica and the Wolfram Language, as described by Wolfram, illustrates the evolution of computational tools that could potentially be used to analyze and visualize the kind of astrophysical data we deal with and we have every credential, exponential. The expression evaluation that relates to the algorithms and methods used has a lot of duck-like behavior, to describe what would happen. Sure we can discuss the propagation of INTEGRAL/SPI data but when I saw your specific empirical data anlysis that consrains dark matter properties, I sort of jumped a bit because I don't have advice, I have a suggestion. Thank you for the Resource function GraphMerge
.
CornerDockingRules[x_, y_, dockingIndex_] :=
Join[{x*y -> dockingIndex},
Table[i -> Subscript[v, i], {i, x*y - 1}]];
Create2DGridGraphCorner[x_, y_, dockingIndex_] :=
Graph[EdgeList[GridGraph[{x, y}]] /.
CornerDockingRules[x, y, dockingIndex]];
FaceDockingRules[x_, y_, dockingIndices_] := Join[
Thread[Table[i, {i, Length[dockingIndices]}] -> dockingIndices],
Table[i -> Subscript[v, i],
{i,
DeleteCases[VertexList[GridGraph[{x, y}]],
Alternatives @@ dockingIndices]
}]];
Create2DGridGraphFace[x_, y_, dockingIndices_] :=
Graph[EdgeList[GridGraph[{x, y}]] /.
FaceDockingRules[x, y, dockingIndices]];
AppendLowerDimensionGridGraph[highDimensionGridGraph_,
lowDimensionGridGraph_] :=
GraphUnion[highDimensionGridGraph, lowDimensionGridGraph];
highDimensionGridGraph = GridGraph[{7, 7, 7},
VertexSize -> 0.3,
VertexStyle -> LightBlue,
EdgeStyle -> Lighter@Gray];
cornerGraph = Create2DGridGraphCorner[15, 15, 1];
faceGraph = Create2DGridGraphFace[15, 15,
{1, 2, 3, 4, 5}];
outlandishGraph = AppendLowerDimensionGridGraph[
AppendLowerDimensionGridGraph[highDimensionGridGraph,
cornerGraph],
faceGraph];
layouts = {"CircularMultipartiteEmbedding",
"LayeredDigraphEmbedding",
"HighDimensionalEmbedding", "SpringElectricalEmbedding",
"SpringEmbedding", "SpectralEmbedding",};
graphs = Table[
Graph[outlandishGraph,
GraphLayout -> layout],
{layout, layouts}];
Grid[Partition[graphs, 2]]
Just remember. You, can do. Whatever, you want, to do. My physical parameters, like lots of rays of light coming in to that lens will make those rays of light converge..so when we look at things far away from a distance, the way those things get focused on our retina and our eye, it does lensing to make light be "bent" and converge together. The same thing happens with light going around a galaxy. The raw observational data that expands our understanding across the universe with data from telescopes and that is your theory of dark matter, your energy distribution in the universe. It's their propagation through the spaceline when neutrinos oscillate that could be used to model the influence of dark matter and dark energy, on photon propagation. To observe the effect of fractional dimensionality we can see the chain of the dimensions and the corner and face docking.
I think that that "time evolution" of a system is usually described by the SchrÃ¶dinger equation, which can involve a Laplacian operator when dealing with spatial variables. In a vacuum, light goes at the standard speed of light but in a piece of glass, light goes one and a half times slower than it does in the vacuum.. so it's going faster ...than the speed of light as it existed in a sheet of glass and when you have the Hamiltonian, that dictates how the wavefunction of a system evolves over time, or the "amplitudes" in quantum mechanics that describe my probabilistic calculation of assigning amplitudes to graph vertices and performing time evolution in the eigenspace of the Laplacian. @Gianmarco Morbelli, it's been fun.
SierGraph[n_, d_] :=
MeshConnectivityGraph[SierpinskiMesh[n, d], 0,
VertexLabels -> Automatic];
fSier[g_, dockingIndex_, vertexname_] :=
Join[{{0, 1} -> dockingIndex},
Table[{0, i} -> Subscript[vertexname, i], {i,
Length[VertexList[g]]}]];
CreateSierpinskiCorner[n_, d_, dockingIndex_, vertexname_] :=
Graph[EdgeList[SierGraph[n, d]] /.
fSier[SierGraph[n, d], dockingIndex, vertexname]];
sierpinskiCorner = CreateSierpinskiCorner[3, 2, 1, a];
gg = GridGraph[{5, 5}];
g1 = Graph[GraphUnion[gg, CreateSierpinskiCorner[3, 2, 1, b]]];
g2 = Graph[GraphUnion[gg, CreateSierpinskiCorner[4, 2, 1, c]]];
g3 = Graph[GraphUnion[gg, CreateSierpinskiCorner[5, 2, 1, d]]];
g4 = Graph[GraphUnion[gg, CreateSierpinskiCorner[6, 2, 1, e]]];
g5 = Graph[GraphUnion[gg, CreateSierpinskiCorner[7, 2, 1, f]]];
g6 = Graph[GraphUnion[gg, CreateSierpinskiCorner[3, 2, 1, g]]];
g7 = Graph[GraphUnion[g6, CreateSierpinskiCorner[4, 2, 25, h]]];
g8 = Graph[GraphUnion[g7, CreateSierpinskiCorner[5, 2, 25, j]]];
g9 = Graph[GraphUnion[g8, CreateSierpinskiCorner[6, 2, 25, k]]];
g10 = Graph[GraphUnion[g8, CreateSierpinskiCorner[7, 2, 25, l]]];
graphs = {g1, g2, g3, g4, g5, g6, g7, g8, g9, g10}
One of the things that really mystified people in the 1800s was what heat is. So when you model the behavior of a classical photon as an electromagnetic (EM) wave through spaces of varying dimensions, there's nothing that I can focus on other than how the intensity of a wave, the caloric fluid flows from the hot metal to the cold metal, the whole wave equation on graphs where we've already established the Laplacian Matrix for integer dimensions and then we can discretize the solution of the wave equation in continuous space and sample it on the vertices of the graph. That is how we model fractional dimensions. It was amazing chaining graphs of different dimensions together because that's how we get them chained, @Simon Fischer it's really about different methods of introducing fractional dimensions to a graph, with particles. The associated production of strange particles, quarks, you know.
The Laplacian Matrix method, this method, doesn't inherently capture fractional dimensions in the operator itself. Space is made of molecules bouncing around, hitting each other..the solution to the wave equation in fractional dimensions involves a function of the dimension, a radial coordinate, and the Hankel function of the second type. And with this solution we built this whole model of physics that seems to work remarkably well, that is based on this idea of a discrete space and these rules being applied. And if you ask, you know what it is, what merges..are there features of space that we're not taking into account? This solution exhibits different spatial behaviors compared to integer dimensions, specifically amplifying or damping the wave amplitude..based on whether the dimension is less than or greater than two. The introduction of curvature into a graph to simulate dimension change can be achieved by adding edges between non-adjacent vertices.
gridToGraph[gridSize_, {x_, y_}] :=
With[{theta = 2 Pi x/gridSize, phi = Pi y/gridSize},
{Cos[theta] Sin[phi], Sin[theta] Sin[phi], Cos[phi]}];
fPertD[g_, vertexname_, pertRadius_] :=
Table[i -> Subscript[vertexname, i],
{i, VertexList[
NeighborhoodGraph[g, First[GraphCenter[g]], pertRadius]]}];
introducePert[g_, pertRadius_, vertexname_] := Graph[EdgeList[g] /.
fPertD[g, vertexname, pertRadius]];
GG1010 = GridGraph[{10, 10}];
vertexCoordinates = gridToSpherical[10, #] & /@
(GraphEmbedding[GG1010]);
graph = Graph[VertexList[GG1010],
EdgeList[GG1010],
VertexCoordinates -> vertexCoordinates,
VertexSize -> 0.1,
EdgeStyle -> Directive[Thickness[0.005]],
ImageSize -> 500];
perturbedGraph = introducePert[graph, 2, a];
perturbedVertexCoordinates =
Thread[VertexList[perturbedGraph] -> gridToGraph[10, #] & /@
GraphEmbedding[perturbedGraph]];
DirectedGraph[
Graph[
perturbedGraph,
VertexCoordinates -> perturbedVertexCoordinates,
VertexLabels -> "Name",
VertexSize -> Thread[VertexList[perturbedGraph] -> Table[
If[IntegerQ[i],
0.1,
0.2],
{i, VertexList[perturbedGraph]}]],
VertexStyle -> Thread[VertexList[perturbedGraph] -> Table[
If[IntegerQ[i],
RGBColor[0.4, 0.8, 1],
RGBColor[1, 0.6, 0.8]],
{i, VertexList[perturbedGraph]}]],
EdgeStyle -> Directive[Thickness[0.001]],
ImageSize -> 500]]
Graphics3D[{
Opacity[0.5],
RGBColor[0.4, 1, 0.6],
Line[{{0, 0, 0}, #}] & /@
(perturbedVertexCoordinates[[All, 2]])}]
It turns out that dark matter ironically is a feature of the microscopic structure of space, just like heat turned out to be a feature of the microscopic structure of materials, the kind of multiway mobile automata that we've been having in our models of physics..our research has generated jealousy in the Wolfram Community. There are so many inquiries and it's full of the effects of dark matter, dark energy, and other cosmological entities, applied to areas such as the study of cosmological structures like black holes and dark matter, in our exploration if you know the formula for the volume of a sphere it's 4/3rds Pi R^3. And formulating gravity as being associated with the curvature of space, you can think about gravity as a feature of small dimension change of space, the "value system" that brings about this introduction of fractional dimensionality within a graph. And that's still up in the air because sometimes, when we get these graphs all solved out it's like introducing curved graphs, the propagation of an electromagnetic wave on a 2D grid graph, with a light flash originating from the center of a sphere.
It's like if you're on a sphere for example and you say let's keep going..at worst I come back around the sphere but I never reach the edge of the sphere that's not a concept that is a thing in the surface of a sphere. In the beginning of the universe, there was; you know there was a Big Bang, it was like using graph theory to model physical spaces, especially those with non-integer dimensions. The universe will start coming back and there will be a big time crunch, the computational simulation of complex physical phenomena that will explain how we crunch time by representing space as a network of connected nodes (vertices), where researchers can explore the behavior of waves as they travel through different structures. But what about people like us who don't even propagate any waves in fractal dimensions? I saw the differential spatial spread sound in the center of the wave, and in its wake it's like how proper time is not a local nor global concept; it is rather associated to a specific reference frame (a priori of arbitrary size).
Now we don't know that matter with a negative mass exists, people say oh we don't really know matter..what is a tensor? All of the implications for Quantum Mechanics and General Relativity for the last hundred years, it's not been possible to have a situation where there's truly nothing there where there's truly a "vacuum" with nothing there. In Quantum Mechanics one of the features of it is to say you never know exactly what is going on. These structures, the fractal dimensions such as those present in the SierpiÅ„ski graph - these structures do not conform to traditional Euclidean geometry and can model naturally bubbling fractal patterns found in various scales of physics, from microscopic to cosmic structures. If the concept of fractional dimensions can be integrated into the standard model of quantum mechanics and general relativity..those atoms of space are constantly being rewritten and offer new ways to understand phenomena like quantum entanglement or the fabric of spacetime near singularities such as black holes. It knits together the structure of space, the activity of the universe does. This is so observer-centric. But that's just the graph. @Simon Fischer.
Plot[{
(Re[x^(1 - (1.585)/2) HankelH2[3, x]] + Re[HankelH2[3, x]])*
Sin[2*Pi*x],
(Re[x^(1 - (2.3)/2) HankelH2[3, x]] + Re[HankelH2[3, x]])*
Sin[2*Pi*x]},
{x, 0, 10},
PlotStyle -> {
Directive[RGBColor[0.4, 0.8, 1], Thick],
Directive[RGBColor[1, 0.6, 0.3], Thick]
},
ImageSize -> Large,
AxesLabel -> {"Radial distance r [a.u.]", "Amplitude [V/m]"},
PlotLegends -> "Expressions"]
Everything that we care about is this tiny piece of fluff, it might be one in ten to the hundred and twenty..one path in a trillion trillion trillion trillion trillion trillion simulations and visualizations of complex wave dynamics and so our embarrassing techniques, are associated with a certain amount of mass. Our vacuum fluctuations, it's sort of like conceptualizing interactions that are difficult to observe directly. The universe isn't curled into a tiny ball but it's cool how if you cross your eyes at these complex systems it sort of kind of looks like..when we lay down and advance in fields that require non-traditional geometric and algebraic frameworks. I'm always down for further exploration into how these concepts can be applied beyond electromagnetism..such as those places where there's less activity that will appear to have less mass..such as in acoustic waves or even gravitational waves. It's a big hack to do with current physics and it only makes us look bad. When you see what happened with the compact-closed dagger symmetric monoidal categories..we really contributed to larger theoretical frameworks theorizing like a champ, every 30 minutes we found fundamental theories of physics through computational exploration. A very sad day for C, the lower level programming language instead of the notation, the starry way that we collaborate on methodological approaches to studying physics. Discretized solutions, the use of and the Laplacian Matrix for modeling wave propagation offers a blueprint for researches like us who are looking, to tackle similar problems in physics or related fields. So we took all of space and we said we're putting down all these angles in different places.
@Simon Fischer? You can say electromagnetism is associated with an arbitrary choice of what amounts of an angle. In electromagnetism, there's this idea of a voltage. We say, in standard electricity we say, we say such and such is a voltage. When you have a 9 volt battery, the difference in voltage is like a man modeling wave propagation between two poles of the battery, volts is a measure of, you can think of it as a measure of electric potential and you can say there's a 9 volt difference between the positive pole and that voltage difference is what pushes electrons through a wire, makes electric currents. But the choice of what, we say one side of the battery it's at 712 volts. The only thing we know is the other side of the battery is 9 volts different from that.
Getamplitudes[g_] := AnnotationValue[g, VertexWeight];
Attributes[Setamplitudes] = {HoldFirst};
Setamplitudes[g_, amplitudes_] :=
AnnotationValue[g, VertexWeight] = amplitudes;
GraphDistSmooth[g_, v1_, v2_] :=
If[String[v1] === String[v2], 1, GraphDistance[g, v1, v2]];
d[g_, v_, perturbedDimension_] :=
If[IntegerQ[v], 2, perturbedDimension];
Phi[g_, vCenter_, v2_, t_, v_, beta_, perturbedDimension_] :=
2*GraphDistSmooth[g, vCenter,
v2]^(1 - (d[g, v2, perturbedDimension])/2) HankelH2[v,
beta*GraphDistSmooth[g, vCenter, v2]]*Cos[t];
newVertexValuesFracDim[g_, v1_, t_, q_, c_, perturbedDimension_] :=
Table[Re[Phi[g, v1, it, t, q, c, perturbedDimension]],
{it, VertexList[g]}];
AmplitudeMatrix3D[g_, n_] := ArrayReshape[Getamplitudes[g], {n, n, n}];
SnowconeColor[z_] := Blend[{RGBColor[0, 0, 1],
RGBColor[1, 1, 1],
RGBColor[1, 0.4, 0.7]}, z];
GG101010 = GridGraph[{10, 10, 10},
VertexShapeFunction -> "Circle",
VertexSize -> Large];
animation3DTableGG = Table[Setamplitudes[GG101010,
newVertexValuesFracDim[GG101010,
First[GraphCenter[GG101010]],
t, 0.3, 1, 3]];
ListPlot3D[AmplitudeMatrix3D[GG101010, 10],
ColorFunction -> SnowconeColor,
PlotRange -> {{1, 10}, {1, 10}, {-2, 2}},
BoxRatios -> {1, 1, 1}], {t, 0, 7, 0.5}];
ListAnimate[animation3DTableGG]
The graph formats..maybe there are other ways to render them. The only thing that means something is the volt difference. So at every point in space you could say there's a voltage, but all that matters is the difference of voltages between different places in space. How does one place in space kind of, if you model wave propagation, how do you relate the voltages effectively in different places in space and one way to think about that is, let's imagine that you have this voltage arranged in space and suddenly you change, I should say, when there's a difference in voltage you can represent that difference by an electric field, and you can kind of say "roughly" what's going to happen is you've got one place where there's a voltage another place where there's a voltage and an electric field that produces a change in one of these changes, that's got to change the electric field. When we change the voltage here that means we make that change, somewhere - how fast does the electric field change?
10 miles away, it doesn't do it instantaneously, it does it at the speed of light. It does it as kind of a wave of change that propagates out, that wave of change is essentially an electromagnetic wave. And the fact that there has to be this wave of change, is the reason that photons exist in this picture. You think of it as a carrier of change, tell the rest of the universe that there was a change in this voltage. So it was a necessary feature of this idea that there was a voltage and this voltage was arbitrary, the difference between gauge and different places has a different effect..as you make a change in gauge, how does the rest of the universe get to know about that change? Gauge fields, like photons, kind of carriers of that change. And the answer is..exploring the propagation of waves through such systems could yield insights into quantum superposition and parallelism and when you say there's a change in this electric potential, well - okay you have to include this change in conventions..the bottom line is this connection field, this carrier of change is something like a photon. Maybe we can render them on Android. Like how we found common ground between us, was really foundational.
With[{gg = GridGraph[{10, 10, 10}, VertexShapeFunction -> "Circle"]},
perturbedGraph3D =
Graph[introducePert[gg, 3, a],
VertexSize ->
Thread[VertexList[introducePert[gg, 3, a]] ->
Table[If[IntegerQ[i], 3, 5], {i,
VertexList[introducePert[gg, 3, a]]}]]]];
newVertexValuesFracDim[g_, v1_, t_, q_, c_, perturbedDimension_] :=
Table[Re[Phi[g, v1, it, t, q, c, perturbedDimension]],
{it, VertexList[g]}];
animation3DTableGGpertD = Table[
Setamplitudes[perturbedGraph3D,
newVertexValuesFracDim[perturbedGraph3D,
First[GraphCenter[perturbedGraph3D]],
t, 0.3, 1, 2.4]];
ListPlot3D[
AmplitudeMatrix3D[perturbedGraph3D, 10],
ColorFunction -> SnowconeColor,
PlotRange -> {{1, 10}, {1, 10}, {-1, 1.5}},
BoxRatios -> {1, 1, 1}],
{t, Table[i, {i, 0, 6.5, 0.5}]}];
ListAnimate[
Rasterize[#, RasterSize -> 400] & /@ animation3DTableGGpertD]
And you're a free man. This rate of change goes at the speed of light, has mass like a photon. Your work relates to multiway systems, a concept from the Wolfram Physics Project that suggests multiple simultaneous states or histories of a system. There's a mathematical theory of zero-mass gauge features..very elegant. Exploring the propagation of waves through systems as such kind of could yield insights into quantum superposition and parallelism, because that's how we understand how photons propagate through spaces of varying dimensions with direct implications for photonics, what we'll call the Higgs field. It interacts with this background thing it's just going happily along but it's being kicked by the fact that there's a background field. Normally, you just have electrons and photons that interact with each other. I'm hanging out around the Higgs field and there are Higgs particles everywhere, the whole universe is densely filled with Higgs particles. It will keep on, it will be keeping on affecting the process of being, where the electron given an effective mass in fractional dimensions, where the Higgs condensate, will make the electron have a mass. So you went from this thing. Thank you for this. So now you know the analytic solution for the wave equation, in the order that it happens.
SierGraph[n_, d_] :=
MeshConnectivityGraph[SierpinskiMesh[n, d], 0,
VertexLabels -> Automatic];
fSier[g_, dockingIndex_, vertexname_] := Join[{{0, 1} -> dockingIndex},
Table[{0, i} -> Subscript[vertexname, i],
{i, Length[VertexList[g]]}]];
CreateSierpCorner[n_, d_, dockingIndex_, vertexname_] := Graph[
EdgeList[SierGraph[n, d]] /.
fSier[SierGraph[n, d],
dockingIndex,
vertexname]];
TableOfSierps[sierpx_, sierpDim_, firstDockingIndex_,
lastDockingIndex_] := Table[
CreateSierpCorner[sierpx, sierpDim, a,
ToString[Part[Alphabet[], a - firstDockingIndex + 1]] <>
ToString[firstDockingIndex]],
{a, firstDockingIndex, lastDockingIndex}];
Getamplitudes[g_] := AnnotationValue[g, VertexWeight];
Attributes[Setamplitudes] = {HoldFirst};
Setamplitudes[g_, amplitudes_] :=
AnnotationValue[g, VertexWeight] = amplitudes;
ColorByamplitude[g_, amplitudes_] :=
HighlightGraph[g,
MapThread[
Style, {VertexList[g], Map[ColorData["Rainbow"], amplitudes]}]];
ColorByInternalamplitude[g_] :=
ColorByamplitude[g, Getamplitudes[g]];
UnnormalizedLaplacian[g_] :=
N@(DiagonalMatrix[VertexOutDegree[#]] - AdjacencyMatrix[#]) &@
UndirectedGraph[g];
GraphSpectrum[
g_] := <|"eigenvalues" -> Eigenvalues[#],
"eigenvectors" -> Eigenvectors[#]|> &@
UnnormalizedLaplacian[g];
EigenSpaceProject[ic_, eigenvectors_] := eigenvectors . ic;
VertexSpaceProject[vector_, eigenvectors_] :=
Transpose[eigenvectors] . vector;
EigenSpaceAdvance[spectrum_, coordinates_,
t_] := (Cos[1/10*t*spectrum["eigenvalues"]] +
Sin[1/10*t*spectrum["eigenvalues"]])*coordinates;
amplitudeSpaceAdvance[spectrum_, ic_, t_] := VertexSpaceProject[
EigenSpaceAdvance[spectrum, EigenSpaceProject[ic, #], t], #] &@
spectrum["eigenvectors"];
Attributes[TimePropagate] = {HoldFirst};
TimePropagate[g_, t_] :=
Setamplitudes[g,
amplitudeSpaceAdvance[GraphSpectrum[g], Getamplitudes[g], t]];
SetDeltaFunction[g_, vertex_, distance_] := Module[{ng = g},
(AnnotationValue[{ng, #}, VertexWeight] =
ConstantArray[1, Length[#]]) &@
VertexList[NeighborhoodGraph[g, vertex, distance]];
ng];
ThreeDGG = GridGraph[{10, 10, 10},
VertexSize -> 6,
VertexShapeFunction -> "Circle",
ImageSize -> 400, VertexLabels -> Automatic];
t1 = TableOfSierps[3, 2, 1, 10];
t2 = TableOfSierps[3, 2, 91, 100];
finalChained3D =
Graph[GraphUnion[GridGraph[{10, 10, 10}], t1[[5]], t2[[5]]],
ImageSize -> 400,
VertexShapeFunction -> "Circle",
VertexSize -> 6];
Setamplitudes[finalChained3D,
ConstantArray[0, Length[VertexList[finalChained3D]]]];
ColorByInternalamplitude[finalChained3D];
deltaGridGraph3DSierp = SetDeltaFunction[finalChained3D,
First[GraphCenter[GridGraph[{10, 10, 10}]]], 1];
ColorByInternalamplitude[deltaGridGraph3DSierp];
animation3DTableSierp = Table[
TimePropagate[deltaGridGraph3DSierp, t];
ColorByInternalamplitude[deltaGridGraph3DSierp],
{t, Table[i, {i, 0, 30, 1}]}];
ListAnimate[
Rasterize[#, RasterSize -> 400] & /@ animation3DTableSierp]
What we focus on is adding our own edges, which might explain the spatial behavior of the wave in cases where the dimension is less than or greater than two. And this thing effectively has a non-zero mass, that's all good, and you have to assume there's a self-interaction with the Higgs particles, if you have particles in a water the molecules of water interact with each other and that's how you get liquid water as opposed to the molecules flowing around and not paying any attention to each other, this condensate together is a consequence of the fact that they have interactions. There's this assumption that the Higgs field has this self-interaction that enriches our understanding of space-time geometry, I always thought this black hole massive object was just a massive hack and eventually, in the Large Hadron Collider, Higgs particles were discovered. 120 times the mass of a proton roughly and in the context of curved space-time around black holes, masses of all the elementary particles. I could quote you all those masses that existed at the time.
But, I am modeling, we know the transmission of signals in the irregular and fractal-like medium of interstellar space, especially when you accompany it with visualizations and simulations..it's an excitation, it's like the eddy in the fluid whereas the underlying fluid is like the whole fluid itself. In that sense we can switch places from the 3D to the Simon Fischer 2D grid graph, this is the place to be.
simpleGridGraph = GridGraph[{10, 10, 10},
ImageSize -> 400,
VertexSize -> 6,
VertexShapeFunction -> "Circle"];
simpleGridGraph = EdgeAdd[simpleGridGraph,
{529 <-> 500, 529 <-> 498, 529 <-> 558, 529 <-> 560, 529 <-> 561,
529 <-> 501, 529 <-> 497, 529 <-> 557, 529 <-> 590, 529 <-> 588,
529 <-> 620, 529 <-> 618, 529 <-> 470, 529 <-> 440, 529 <-> 468,
529 <-> 438}];
Setamplitudes[simpleGridGraph, ConstantArray[0, 1000]];
ColorByInternalamplitude[simpleGridGraph];
deltaGridGraphLocalPert = SetDeltaFunction[simpleGridGraph, 259, 1];
ColorByInternalamplitude[deltaGridGraphLocalPert];
TimePropagate[deltaGridGraphLocalPert, 0];
ColorByInternalamplitude[deltaGridGraphLocalPert];
animation3DTableLocalPert =
Table[TimePropagate[deltaGridGraphLocalPert, t];
ColorByInternalamplitude[deltaGridGraphLocalPert],
{t, Table[i, {i, 0, 16, 0.5}]}];
ListAnimate[Rasterize[#, RasterSize -> 400] & /@
animation3DTableLocalPert]
I just show that the intensity of the wave gets amplified or damped whatever the dimensional shape. With these high-energy particles..we've got these electron-positron pairs, charge -1 and +1, and there's a cloud of electrons and positrons around the electron which in some sense will screen the charge of the electron, roughly.
edAddFct[g_] :=
Sort /@ UndirectedEdge @@@
Position[Outer[EuclideanDistance@## &, #, #, 1], N@Sqrt@3] &@
GraphEmbedding@g // Union
AddCrossBars[g_] := EdgeAdd[g, edAddFct[g]]
simpleCrossGraph = EdgeAdd[GridGraph[{10, 10, 10},
ImageSize -> 400,
VertexSize -> 6,
VertexShapeFunction -> "Circle"],
edAddFct[GridGraph[{10, 10, 5}]]];
Setamplitudes[simpleCrossGraph,
ConstantArray[0, Length[VertexList[simpleCrossGraph]]]];
ColorByInternalamplitude[simpleCrossGraph];
deltaGGCrossBar = SetDeltaFunction[simpleCrossGraph, 500, 1];
ColorByInternalamplitude[simpleCrossGraph];
animation3DTableCross = Table[TimePropagate[deltaGGCrossBar, t];
ColorByInternalamplitude[deltaGGCrossBar],
{t, Table[i, {i, 0, 16, 0.5}]}];
ListAnimate[animation3DTableCross]
These are quite some models for fractional dimensional shape.
Fractional dimensions can cause damping or amplifying effects on the amplitude of EM waves, exploring the concept of curvature and modeling non-integer, spatial dimensions..this graph is particularly electromagnetically heating up, to say the least. Less than a microsecond after the beginning of the universe one wouldn't expect this condensate to exist. This condensate has a bunch of mass associated with it. So that means the fertile intersection of graph theory, physics, and computational modeling attracts scholars at us at the cafÃ©, at the library meeting up for a chess match from diverse fields where we can contribute to a more holistic understanding of the universe's fabric, it's not like we can't efficiently handle the computations involved with this blanket notion that we can model wave propagation in variable-dimensional spaces! It would have to be as if one took steam and made it colder and colder and still had to turn it into water by a factor of a billion. I see the Simon Fischer resource functions for heterodimensional graphs, arbitrary wave equations, and grid graphs with crossbars. Yeah, I would like to express gratitude to you Simon for what happened between us and for supervising part of my project, with support and encouragement from Stephen Wolfram. The photon can carry momentum.
simpleGridGraph =
GridGraph[{16, 16}, VertexSize -> Large, ImageSize -> 350];
AnnotationValue[simpleGridGraph, VertexWeight];
Getamplitudes[g_] := AnnotationValue[g, VertexWeight];
Attributes[Setamplitudes] = {HoldFirst};
Setamplitudes[g_, amplitudes_] :=
AnnotationValue[g, VertexWeight] = amplitudes;
ColorByamplitude[g_, amplitudes_] :=
HighlightGraph[g,
MapThread[
Style, {VertexList[g], Map[ColorData["Rainbow"], amplitudes]}]];
ColorByInternalamplitude[g_] :=
ColorByamplitude[g, Getamplitudes[g]];
SetDeltaFunction[g_, vertex_, distance_] :=
Module[{ng =
g}, (AnnotationValue[{ng, #}, VertexWeight] =
ConstantArray[1, Length[#]]) &@
VertexList[NeighborhoodGraph[g, vertex, distance]]; ng];
UnnormalizedLaplacian[g_] :=
N@(DiagonalMatrix[VertexOutDegree[#]] - AdjacencyMatrix[#]) &@
UndirectedGraph[g];
AdjMatrix[g_] := N@AdjacencyMatrix[#] &@UndirectedGraph[g];
GraphSpectrum[
g_] := <|"eigenvalues" -> Eigenvalues[#],
"eigenvectors" -> Eigenvectors[#]|> &@UnnormalizedLaplacian[g];
EigenSpaceProject[ic_, eigenvectors_] := eigenvectors . ic;
VertexSpaceProject[vector_, eigenvectors_] :=
Transpose[eigenvectors] . vector;
EigenSpaceAdvance[spectrum_, coordinates_,
t_] := (Cos[1/10*t*spectrum["eigenvalues"]] +
Sin[1/10*t*spectrum["eigenvalues"]])*coordinates;
amplitudeSpaceAdvance[spectrum_, ic_, t_] :=
VertexSpaceProject[
EigenSpaceAdvance[spectrum, EigenSpaceProject[ic, #], t], #] &@
spectrum["eigenvectors"];
Attributes[TimePropagate] = {HoldFirst};
TimePropagate[g_, t_] :=
Setamplitudes[g,
amplitudeSpaceAdvance[GraphSpectrum[g], Getamplitudes[g], t]];
Setamplitudes[simpleGridGraph, ConstantArray[0, 256]];
ColorByInternalamplitude[simpleGridGraph];
deltaGridGraph =
SetDeltaFunction[simpleGridGraph,
First[GraphCenter[simpleGridGraph]], 1];
ColorByInternalamplitude[deltaGridGraph];
animation2DTable =
Table[TimePropagate[deltaGridGraph, t];
ColorByInternalamplitude[deltaGridGraph], {t,
Table[i, {i, 1, 10, 0.5}]}];
ListAnimate[animation2DTable]
But later on, people said look it's such a cool effect that you get this exponential expansion that something must work out okay with respect to the supercooling phenomenon, it led to a lot of scenarios that are patched and so on; I think the story of the universe and its expansion are that the universe started out infinite-dimensional and "required" all these patches in existing theories, which could benefit a wide range of computational applications. The potential of mass density, non-Euclidean geometries in physics, suggests ways these geometries could be employed in practical applications such as the expansion of the universe that accelerates; relative to us at some particular place in the universe, there are parts of the universe that are receding from us faster than the speed of light, the universe tears itself apart and there are pieces we just don't know what's going to happen to them because you might be sending, exchanging radio signals, the radio signals will never get to you.
EigenSpaceAdvance[spectrum_, coordinates_,
t_] := (Cos[1/10*t*spectrum["eigenvalues"]])*coordinates;
Setamplitudes[simpleGridGraph, ConstantArray[0, 256]];
ColorByInternalamplitude[simpleGridGraph];
deltaGridGraph =
SetDeltaFunction[simpleGridGraph,
First[GraphCenter[simpleGridGraph]], 1];
ColorByInternalamplitude[deltaGridGraph];
animation2DTable = Table[TimePropagate[deltaGridGraph, t];
Graph[ColorByInternalamplitude[deltaGridGraph],
VertexSize ->
Thread[VertexList[ColorByInternalamplitude[deltaGridGraph]] ->
0.8*(Abs[
AnnotationValue[ColorByInternalamplitude[deltaGridGraph],
VertexWeight]])]], {t, Table[i, {i, 0, 8, 0.5}]}];
ListAnimate[animation2DTable]
Inside a black hole, that radio signal will never get out and it's the same type of thing here because of this thing called Principle of Equivalence, the equivalence between gravity and acceleration you will just never get, this signal so to speak so I think that by intertwining theoretical physics with computational and mathematical techniques we get this erratic, irregular and fractal-like medium of interstellar space where we can transmit these electromagnetic waves, photon propagation through variable-dimensional space.
f[x_, y_, dockingIndex_] :=
Join[{x*y -> dockingIndex}, Table[i -> Subscript[v, i], {i, x*y - 1}]]
Create2DGGCorner[x_, y_, dockingIndex_] :=
Graph[EdgeList[GridGraph[{x, y}]] /. f[x, y, dockingIndex]]
fFace[x_, y_, dockingIndices_] :=
Join[Thread[Table[i, {i, Length[dockingIndices]}] -> dockingIndices],
Table[i -> Subscript[v, i], {i,
DeleteCases[VertexList[GridGraph[{x, y}]],
Alternatives @@ Table[i, {i, Length[dockingIndices]}]]}]]
Create2DGGFace[x_, y_, dockingIndices_] :=
Graph[EdgeList[GridGraph[{x, y}]] /. fFace[x, y, dockingIndices]]
appendLowerDGG[highDGG_, lowDGG_] := GraphUnion[highDGG, lowDGG]
appendLowerDGG[GridGraph[{5, 5, 5}], Create2DGGCorner[5, 5, 1]]
appendLowerDGG[GridGraph[{5, 5, 5}],
Create2DGGFace[5, 5, {1, 2, 3, 4, 5}]]
SierGraph[n_, d_] :=
MeshConnectivityGraph[SierpinskiMesh[n, d], 0,
VertexLabels -> Automatic];
SierGraph[2, 3]
fSier[g_, dockingIndex_, vertexname_] :=
Join[{{0, 1} -> dockingIndex},
Table[{0, i} -> Subscript[vertexname, i], {i,
Length[VertexList[g]]}]]
CreateSierpCorner[n_, d_, dockingIndex_, vertexname_] :=
Graph[EdgeList[SierGraph[n, d]] /.
fSier[SierGraph[n, d], dockingIndex, vertexname]]
Graph[GraphUnion[GridGraph[{5, 5}], CreateSierpCorner[3, 2, 1, v]],
ImageSize -> 500]
fPertD[g_, vertexname_, pertRadius_] :=
Table[i -> Subscript[vertexname, i], {i,
VertexList[
NeighborhoodGraph[g, First[GraphCenter[g]], pertRadius]]}]
introducePert[g_, pertRadius_, vertexname_] :=
Graph[EdgeList[g] /. fPertD[g, vertexname, pertRadius]]
GG1010 = GridGraph[{10, 10}];
g1 = introducePert[GG1010, 2, v];
perturbedGraph1010 =
Graph[g1,
VertexSize ->
Thread[VertexList[g1] ->
Table[If[IntegerQ[i], 0.2, 0.9], {i, VertexList[g1]}]],
ImageSize -> 300]
Plot[{Re[x^(1 - (1.585)/2) HankelH2[3, 1*x]],
Re[HankelH2[3, 1*x]]}, {x, 0, 100}, PlotLegends -> "Expressions",
ImageSize -> Large,
AxesLabel -> {"Radial distance r [a.u.]", "Amplitude [V/m]"}]
Plot[{Re[x^(1 - (2.3)/2) HankelH2[3, 1*x]], Re[HankelH2[3, 1*x]]}, {x,
0, 100}, PlotLegends -> "Expressions", ImageSize -> Large,
AxesLabel -> {"Radial distance r [a.u.]", "Amplitude [V/m]"}]
Phi[g_, vCenter_, v2_, t_, v_, beta_, perturbedDimension_] :=
2*GraphDistSmooth[g, vCenter,
v2]^(1 - (d[g, v2, perturbedDimension])/2) HankelH2[v,
beta*GraphDistSmooth[g, vCenter, v2]]*Cos[t]
GraphDistSmooth[g_, v1_, v2_] :=
If[String[v1] === String[v2], 1, GraphDistance[g, v1, v2]]
d[g_, v_, perturbedDimension_] :=
If[IntegerQ[v], 2, perturbedDimension]
dEstimate[g_, v_] :=
ResourceFunction["WolframHausdorffDimension"][g, v]
newVertexValuesFracDim[g_, v1_, t_, q_, c_, perturbedDimension_] :=
Table[Re[Phi[g, v1, it, t, q, c, perturbedDimension]], {it,
VertexList[g]}]
GG2020 =
GridGraph[{20, 20}, VertexShapeFunction -> "Circle",
VertexSize -> Large];
animation2DTableGG =
Table[Setamplitudes[GG2020,
newVertexValuesFracDim[GG2020, First[GraphCenter[GG2020]], t, 0.3,
1, 2]];
ColorByInternalamplitude[GG2020], {t, Table[i, {i, 0, 7, 0.5}]}];
ListAnimate[animation2DTableGG]
And I've got energy inside the atom, the photon comes in and excites the atom. It's sort of like playing the piano, the economics of how we can use the Laplacian Matrix for integer dimensions, discretize continuous solutions for fractional dimensions, and explore integer and fractional dimensional spaces, including the use of structures like the SierpiÅ„ski graph to represent fractional dimensions that propels us to look up at these strange fish, and then look down and call these animations by the same name, because we keep calling them by the same name, because the "original" ListAnimate2DTable doesn't even exist and when something doesn't even exist, the dimensional change..depending on the dimensional change (integer or fractional), the intensity of the wave can either amplify or dampen.
fPertD[g_, vertexname_, pertRadius_] :=
Table[i -> Subscript[vertexname, i], {i,
VertexList[
NeighborhoodGraph[g, First[GraphCenter[g]], pertRadius]]}]
introducePert[g_, pertRadius_, vertexname_] :=
Graph[EdgeList[g] /. fPertD[g, vertexname, pertRadius],
ImageSize -> 700, VertexLabels -> None]
GG2020 = GridGraph[{20, 20}, VertexShapeFunction -> "Circle"];
g2 = introducePert[GG2020, 3, v];
perturbedGraph2020 =
Graph[g2,
VertexSize ->
Thread[VertexList[g2] ->
Table[If[IntegerQ[i], 0.6, 0.9], {i, VertexList[g2]}]],
ImageSize -> 400];
perturbedGraph2020GridForm =
Graph[perturbedGraph2020,
VertexCoordinates ->
VertexList[
perturbedGraph2020] /. (Thread[
VertexList[GG2020] -> (GG2020 // GraphEmbedding)] /.
fPertD[GG2020, v, 3])];
animation2DTableGGpertD =
Table[Setamplitudes[perturbedGraph2020GridForm,
newVertexValuesFracDim[perturbedGraph2020GridForm, 83, t, 0.3, 1,
1.585]];
ColorByInternalamplitude[perturbedGraph2020GridForm], {t,
Table[i, {i, 0, 6.5, 0.5}]}];
ListAnimate[
Rasterize[#, RasterSize -> 400] & /@ animation2DTableGGpertD]
This amplification or dampening is a significant finding which has multiple connotations including as a collection, of the things that we want. The interaction between a Higgs field and a particle field, the Electron field is called Psi, wave behavior changes in spaces with dimensions less than, equal to, or greater than. The showcase of the power of computational methods in simulating complex physical phenomena is awesome, and everything links together to what I predicate on these conditions, the graph theory, differential equations, and topology with physical theories and that masquerades as a mass for theoretical exploration as a field.
TwoDGG =
GridGraph[{20, 20}, VertexSize -> 2,
VertexShapeFunction -> "Circle", ImageSize -> 400,
VertexLabels -> Automatic];
TableOfSierps[sierpx_, sierpDim_, firstDockingIndex_,
lastDockingIndex_] :=
Table[CreateSierpCorner[sierpx, sierpDim, a,
ToString[Part[Alphabet[], a - firstDockingIndex + 1]] <>
ToString[firstDockingIndex]], {a, firstDockingIndex,
lastDockingIndex}]
t1 = TableOfSierps[3, 2, 1, 10];
t2 = TableOfSierps[3, 2, 91, 100];
finalChained =
Graph[GraphUnion[GridGraph[{10, 10}], t1[[5]], t2[[5]]],
ImageSize -> 400, VertexShapeFunction -> "Circle",
VertexSize -> 2.5];
Setamplitudes[finalChained,
ConstantArray[0, Length[VertexList[finalChained]]]];
ColorByInternalamplitude[finalChained];
EigenSpaceAdvance[spectrum_, coordinates_,
t_] := (Cos[1/10*t*spectrum["eigenvalues"]] +
Sin[1/10*t*spectrum["eigenvalues"]])*coordinates;
deltaGridGraph2DSierp =
SetDeltaFunction[finalChained,
First[GraphCenter[GridGraph[{10, 10}]]], 1];
ColorByInternalamplitude[deltaGridGraph2DSierp];
animation2DTableSierp =
Table[TimePropagate[deltaGridGraph2DSierp, t];
ColorByInternalamplitude[deltaGridGraph2DSierp], {t,
Table[i, {i, 0, 30, 1}]}];
ListAnimate[
Rasterize[#, RasterSize -> 400] & /@ animation2DTableSierp]
So, let's see advanced optical systems and how they potentially impact fiber optics, laser technology, and quantum computing with regard to material science and how we can perform chemical engineering in front of a mirror, we can understand wave propagation in the sense of various laminated..dimensional spaces that inform the design of new materials with specific optical properties. The understanding that we need of topological stability, how do we need it? The particle interactions in Quantum Field Theory and the topology of a donut with a hole in it, because ha, the understanding that we have of space-time is like a torus.
simpleGridGraph =
GridGraph[{30, 30}, ImageSize -> 400, VertexSize -> 0.6,
VertexShapeFunction -> "Circle"];
Setamplitudes[simpleGridGraph, ConstantArray[0, 900]];
ColorByInternalamplitude[simpleGridGraph];
deltaGridGraphLocalPert = SetDeltaFunction[simpleGridGraph, 259, 1];
ColorByInternalamplitude[deltaGridGraphLocalPert];
TimePropagate[deltaGridGraphLocalPert, 0];
ColorByInternalamplitude[deltaGridGraphLocalPert];
animation2DTableLocalPert =
Table[TimePropagate[deltaGridGraphLocalPert, t];
ColorByInternalamplitude[deltaGridGraphLocalPert], {t,
Table[i, {i, 0, 16, 0.5}]}];
ListAnimate[
Rasterize[#, RasterSize -> 400] & /@ animation2DTableLocalPert]
And astrophysics' signal transmissions in complex interstellar mediums like jam just injected into the center of this..sphere like the geometry around massive cosmic bodies that we rip, blobs of dough that we study and we have to rip the dough, we can't just continuously deform it to go from the no hole to the hole hole situation, which bridges various mathematical concepts like graph theory, and the differential equations that allow us to perform partial navigation of topological spaces in nautical exploration of how fluid has a certain velocity, where we are not, we are discretizing space on a graph and applying different mathematical and computational techniques akin to some kind of skyrmion that you can't get rid of, that's kind of the thing. It was like discretizing a whole new space on a graph and we even got on a line and took a step up and a step down. Where do I get to when I flip a coin, I get all these wiggly curves that come up and down and they wiggle around and first you get 0. Then, you get the average distance about the square root of the number of steps. Then, the overall distribution is called the Gaussian distribution or normal distribution and the Bell curve shows up all over the place. And you get heads, heads, heads, and tails, tails, tails, and every random walk in one dimension eventually gets back to 0 again. And that's a phenomenon that I'm going to regret, let's say I'm ahead and I'm winning and I indirectly touch upon concepts like quantum superposition and parallelism. Gambler's ruin - eventually you're going to have enough tails that you get right back to 0 right again. It's not about you, it's about fair-weather three-dimensional origin return, in the sense of the long interim before we got back to the origin again, there's this specific optical property of static that, every once in a blue moon you get this interactive simulation that paves the way for more collaborative efforts in understanding the universe. If you could make something 2.5 dimensional, what would it be like? It's sort of the first time that we have a robust way of talking about fractional dimensions..and it doesn't quite work generically there so there may very well be something so yes, it is right that you could take a random walk on this graph, how many points do you get to by going a certain distance out? And that is closely related to the question of what's the probability that you come back on the random walk, if you had an equation for heat; the diffusion equation, start a very very large number of random walks, what is the density, what is the Gaussian distribution of the Bell curve, there are fractional dimensions in the return time of random walks that give you a "cloud" of probabilities of coming back to the origin.