In this paper we discuss a specific follow-up result from the work about the 'theory of p-nomial ... more In this paper we discuss a specific follow-up result from the work about the 'theory of p-nomial triangles'. We want to present some key developments. They involve the representation of twelve connected p-nomial triangles in three dimensional space in the form of a Magen David star pyramid. We construct the representation via coefficient [boundary] vectors, before we show that the established Magen David star pyramid is homotopic to the dodecagonal -.
A link between d-dimensional p−nomial simplices and − algebraic topology, 2022
In this paper we discuss specific follow−up results from the upcoming paper about the ’theory of ... more In this paper we discuss specific follow−up results from the upcoming paper about the ’theory of d−dimensional p−nomial simplices’. As they are strongly connected to ’p−nomial combinatorial number theory’ and to ’probability calculus’ we want to present some key developments. These include ’multivariate (inverse) continuous probabilities established via p−nomial combinatorial sums’, topological probability ’Yvon spaces’, which are not only constructed through different kind of −, but also with p − q−adic power series and product topologies, specific recurring ’Lechenault probability structures’, the ’Vasnier elliptic curves in/on p−nomial manifolds’ and the ’Pintenet surgery theory of m−dimensional p−nomial triangles’. As the second part, we add an explanation, since this paper’s scope is to reach a far wider range of society.
In this paper we will connect 'combinatorial number theory' to 'planar geometry' by firstly prese... more In this paper we will connect 'combinatorial number theory' to 'planar geometry' by firstly presenting the original derivation of the circle constant π and some formulas in which it appears, as for example Vieta's − and Wallis product. Then, we will prove that on the circle other constants can be found. Afterwards we compute for some of these their special value and generalize their construction. In the end we will show, that geometric shapes can stand in resonance with each other. Therefore, we introduce for these kind of elements the idea of 'resonance classes', give an example in detail and expand the previous made results to higher dimensions. We also introduce Hendy's identity that connects the square root of two, one and π to each other.
ME −, Bernoulli and Eulers numbers via the generating eta − and zeta functions, 2019
In this paper we analyze the different generating eta − and zeta functions, which in return prod... more In this paper we analyze the different generating eta − and zeta functions, which in return produce the ME numbers that are directly linked to the Bernoulli −. We establish the characteristics of these numbers, their computation, their values and the domain of the eta − and zeta functions in the complex plain by analytic continuation. Further, we derive formulas for specific values of each function. Then we will analyze their connection with the Γ − and the Riemann ζ function. We also take a look at root series and infinite products representations of periodic polynomials.
Symmetry breaking in Ellen's p-nomial d-dimensional simplices, 2019
In this paper, a partial preview of the upcoming theory of p−nomial triangles, we present first t... more In this paper, a partial preview of the upcoming theory of p−nomial triangles, we present first the construction of Ellen’s p−nomial triangles via the elementary addition rule before we come to their generalization as multifaced d−dimensional simplices; at the end we will also present some constructed with the simple product rule. The triangles in question prove in return by demonstration, that p−nomial triangles, respectively simplices exist which have varying [binomial and exponential] distributions underlying. The importance of this result can only be understood fully with the theory of p−nomial triangles. We will further exhibit ’symmetry points’ and ’− breaking points’ in both, the triangles as well as the simplices, marking an important connection between probability calculus, number theory, algebra and algebraic topology. The paper neither includes the Pintenet surgery theory of p−nomial triangles and d−dimensional − simplices, nor the homotopy of the projections of these into R3.
Combinatorial Coordinates and Riemannian Geometry, 2019
In this paper we apply the combinatorial series in order to establish spherical, hyperbolic and e... more In this paper we apply the combinatorial series in order to establish spherical, hyperbolic and elliptic combinatorial coordinates. Afterwards we construct combinatorial line elements, combinatorial operators and a combinatorial metric tensor to see how their composition differs from the Euclidian, Minkowskian and Riemannian line element, (the standard) differential operators and the metric. Then, we implement the combinatorial series into Riemannian geometry, investigate combinatorial gravitational waves, spaces and spheres.
Canonical and the prime canonical combinatorial matrix, 2016
In this paper we want to employ the different applications, especially those of linear algebra, o... more In this paper we want to employ the different applications, especially those of linear algebra, onto our findings of the combinatorial number theory in order to get a better understanding of the Goldbach - and Landau hypothesis. We do that by establishing both, the canonical − and the prime canonical combinatorial matrix, and some of their properties.
In this paper we want to examine the (combinatorial distribution of) prime numbers and (combinato... more In this paper we want to examine the (combinatorial distribution of) prime numbers and (combinatorial) residue classes. In order to create an algorithm, that can actually determine prime numbers, we connect linear algebra to combinatorial number theory.
In this paper we analyse the construction of p−adic numbers, which we expand to the new combinato... more In this paper we analyse the construction of p−adic numbers, which we expand to the new combinatorial p − q−adic numbers. Thus, introducing combinatorics into algebraic number theory. After we validate the made progress, the topology and the algebra, we construct new functions, which are embedded infinite (combinatorial − and or periodic tower of) fields. Before we come to categories and functors, we also introduce briefly p−q−adic m−nomial combinatorial functions, respectively − series and give an example of a distribution function for it. Further, we show how a p−q−adic distribution function might be constructed.
In this paper we expand the prime number theorem, twice. Then we use both expansions to describe ... more In this paper we expand the prime number theorem, twice. Then we use both expansions to describe the distribution of primes. Afterwards we analyze the symmetry in the asymptotic equivalence. Before we come to the Weil conjecture, we derive a solution of the non-trivial zeros of the Riemann zeta function, the root identity, which we expand to the entire complex plane via a replication identity and which we employ to show, that the critical strip next to the critical line is a zero free zone.
In this paper we discuss a specific follow-up result from the work about the 'theory of p-nomial ... more In this paper we discuss a specific follow-up result from the work about the 'theory of p-nomial triangles'. We want to present some key developments. They involve the representation of twelve connected p-nomial triangles in three dimensional space in the form of a Magen David star pyramid. We construct the representation via coefficient [boundary] vectors, before we show that the established Magen David star pyramid is homotopic to the dodecagonal -.
A link between d-dimensional p−nomial simplices and − algebraic topology, 2022
In this paper we discuss specific follow−up results from the upcoming paper about the ’theory of ... more In this paper we discuss specific follow−up results from the upcoming paper about the ’theory of d−dimensional p−nomial simplices’. As they are strongly connected to ’p−nomial combinatorial number theory’ and to ’probability calculus’ we want to present some key developments. These include ’multivariate (inverse) continuous probabilities established via p−nomial combinatorial sums’, topological probability ’Yvon spaces’, which are not only constructed through different kind of −, but also with p − q−adic power series and product topologies, specific recurring ’Lechenault probability structures’, the ’Vasnier elliptic curves in/on p−nomial manifolds’ and the ’Pintenet surgery theory of m−dimensional p−nomial triangles’. As the second part, we add an explanation, since this paper’s scope is to reach a far wider range of society.
In this paper we will connect 'combinatorial number theory' to 'planar geometry' by firstly prese... more In this paper we will connect 'combinatorial number theory' to 'planar geometry' by firstly presenting the original derivation of the circle constant π and some formulas in which it appears, as for example Vieta's − and Wallis product. Then, we will prove that on the circle other constants can be found. Afterwards we compute for some of these their special value and generalize their construction. In the end we will show, that geometric shapes can stand in resonance with each other. Therefore, we introduce for these kind of elements the idea of 'resonance classes', give an example in detail and expand the previous made results to higher dimensions. We also introduce Hendy's identity that connects the square root of two, one and π to each other.
ME −, Bernoulli and Eulers numbers via the generating eta − and zeta functions, 2019
In this paper we analyze the different generating eta − and zeta functions, which in return prod... more In this paper we analyze the different generating eta − and zeta functions, which in return produce the ME numbers that are directly linked to the Bernoulli −. We establish the characteristics of these numbers, their computation, their values and the domain of the eta − and zeta functions in the complex plain by analytic continuation. Further, we derive formulas for specific values of each function. Then we will analyze their connection with the Γ − and the Riemann ζ function. We also take a look at root series and infinite products representations of periodic polynomials.
Symmetry breaking in Ellen's p-nomial d-dimensional simplices, 2019
In this paper, a partial preview of the upcoming theory of p−nomial triangles, we present first t... more In this paper, a partial preview of the upcoming theory of p−nomial triangles, we present first the construction of Ellen’s p−nomial triangles via the elementary addition rule before we come to their generalization as multifaced d−dimensional simplices; at the end we will also present some constructed with the simple product rule. The triangles in question prove in return by demonstration, that p−nomial triangles, respectively simplices exist which have varying [binomial and exponential] distributions underlying. The importance of this result can only be understood fully with the theory of p−nomial triangles. We will further exhibit ’symmetry points’ and ’− breaking points’ in both, the triangles as well as the simplices, marking an important connection between probability calculus, number theory, algebra and algebraic topology. The paper neither includes the Pintenet surgery theory of p−nomial triangles and d−dimensional − simplices, nor the homotopy of the projections of these into R3.
Combinatorial Coordinates and Riemannian Geometry, 2019
In this paper we apply the combinatorial series in order to establish spherical, hyperbolic and e... more In this paper we apply the combinatorial series in order to establish spherical, hyperbolic and elliptic combinatorial coordinates. Afterwards we construct combinatorial line elements, combinatorial operators and a combinatorial metric tensor to see how their composition differs from the Euclidian, Minkowskian and Riemannian line element, (the standard) differential operators and the metric. Then, we implement the combinatorial series into Riemannian geometry, investigate combinatorial gravitational waves, spaces and spheres.
Canonical and the prime canonical combinatorial matrix, 2016
In this paper we want to employ the different applications, especially those of linear algebra, o... more In this paper we want to employ the different applications, especially those of linear algebra, onto our findings of the combinatorial number theory in order to get a better understanding of the Goldbach - and Landau hypothesis. We do that by establishing both, the canonical − and the prime canonical combinatorial matrix, and some of their properties.
In this paper we want to examine the (combinatorial distribution of) prime numbers and (combinato... more In this paper we want to examine the (combinatorial distribution of) prime numbers and (combinatorial) residue classes. In order to create an algorithm, that can actually determine prime numbers, we connect linear algebra to combinatorial number theory.
In this paper we analyse the construction of p−adic numbers, which we expand to the new combinato... more In this paper we analyse the construction of p−adic numbers, which we expand to the new combinatorial p − q−adic numbers. Thus, introducing combinatorics into algebraic number theory. After we validate the made progress, the topology and the algebra, we construct new functions, which are embedded infinite (combinatorial − and or periodic tower of) fields. Before we come to categories and functors, we also introduce briefly p−q−adic m−nomial combinatorial functions, respectively − series and give an example of a distribution function for it. Further, we show how a p−q−adic distribution function might be constructed.
In this paper we expand the prime number theorem, twice. Then we use both expansions to describe ... more In this paper we expand the prime number theorem, twice. Then we use both expansions to describe the distribution of primes. Afterwards we analyze the symmetry in the asymptotic equivalence. Before we come to the Weil conjecture, we derive a solution of the non-trivial zeros of the Riemann zeta function, the root identity, which we expand to the entire complex plane via a replication identity and which we employ to show, that the critical strip next to the critical line is a zero free zone.
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