Related papers: On the Complex Conjugate Series for Pi
Using WZ pairs we present an infinite family of accelerated series for computing $\zeta(3)$.
A multicomplex, also known as a twisted chain complex, has an associated spectral sequence via a filtration of its total complex. We give explicit formulas for all the differentials in this spectral sequence.
Weanalyzethecomputationalcomplexityofanalgorithmtosolve the conjugacy search problem in a certain family of metabelian groups. We prove that in general the time complexity of the conjugacy search problem for these groups is at most…
We present new infinite arctangent sums and infinite sums of products of arctangents. Many previously known evaluations appear as special cases of the general results derived in this paper.
It is shown that if an infinite synchronized system has a flip, then it has infinitely many non-conjugate flips, and that the result cannot be extended to the class of coded systems.
Separation systems are posets with additional structure that form an abstract setting in which tangle-like clusters in graphs, matroids and other combinatorial structures can be expressed and studied. This paper offers some basic theory…
There exist combable groups in which the conjugacy problem is unsolvable. The isomorphism problem is unsolvable for certain recursive sequences of finite presentations of combable groups.
We consider embeddings of a finite complex in a sphere. We give a homotopy theoretic classification of such embeddings in a wide range.
Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular…
We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many…
Using a pointwise version of Fej\'{e}r's theorem about Fourier series, we obtain two formulae related to the series representations of positive integral powers of $\pi$. We also check the correctness of our formulae by the applications of…
An algorithm for computing /pi(N) is presented.It is shown that using a symmetry of natural numbers we can easily compute /pi(N).This method relies on the fact that counting the number of odd composites not exceeding N suffices to calculate…
In the previous work [2] (i.e., arXiv:2105.03385), we considered continuous solutions of an iterative equation involving the multiplication of iterates. In this paper, we continue to investigate this equation for differentiable solutions.…
This article is about Pi Formulas, infinite series of fractions which sum to multiples of Pi. Each such one can be associated with a unique set $S_k$ of rough numbers, where $k$ is a prime number. Given $S_k$ for any prime $k$, the set…
Coclass theory can be used to define infinite families of finite p-groups of a fixed coclass. It is conjectured that the groups in one of these infinite families all have isomorphic mod-p cohomology rings. Here we prove that almost all…
Many invariants of finitely generated positive cancelative commutative semigroups can be studied from their Poincar\'e series. We offer and present several closed formulas for them. Moreover, those formulas have elementary proofs and are…
We present a method to compute integral cohomology of posets. This toolbox is applicable as soon as the sub-posets under each object possess certain structure. This is the case for simplicial complexes and simplex-like posets. The method is…
The finitistic dimension of a triangulated category is introduced. For the category of perfect complexes over a ring it is shown that this dimension is finite if and only if the small finitistic dimension of the ring is finite.
Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…
We obtained a new formula for $\pi$.