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[7] viXra:2110.0129 [pdf] submitted on 2021-10-22 21:11:51
Authors: Gregory M. Sobko
Comments: 13 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]
This is a brief outline of results represented in the Ph.D. dissertation defended by the author in the Department of Mathematics at Moscow State University (MSU), Russia, in 1973. The complete manuscript of the dissertation is not published. Dissertation “Limit Theorems for Random Walks on Differential Manifolds” is dedicated to the generalization of classical limit theorems of Probability Theory, known as Central Limit Theorem, to the case when sequences of ‘random variables’ represent random walks on differentiable manifolds or generated by products of random elements of noncommutative locally compact groups Lie. Classical results of Probability Theory related to the summation of random variables provide a description of the family of limit laws for distribution of sums of uniformly distributed infinitesimal independent terms and establish conditions for convergence of the sums to the laws of the family. Problems related to probability distributions for sums of “small “ random variables were generalized in two major directions: first, it had been stated a more general problem to move from discrete random walks to Markov processes with continuous time (A.N. Kolmogorov [1], A. Y. Khintchin {2], A.V. Skorokhod [3]); second, the studies shifted to the situation in which the random variables are taking their values on sets of more general mathematical nature than n-dimensional Euclidean space, for example, differentiable manifolds and Lie groups (A. N. Kolmogorov [4], K. Ito [5], G.A. Hunt [6], D. When [7]. In our study, we consider sequences of random walks (i.e. random processes with discrete time) of both Markov and non-Markov character on locally-compact differentiable manifolds, and the limit transition from the walks to Markov random processes with continuous time so that the “steps” of walks are asymptotically uniformly small, and the number of “steps” goes to infinity. The mentioned Markov processes (diffusion and stochastically continuous without the second type of discontinuities) are introduced “constructively”, which provides the information about the properties of transition probabilities, necessary for proofs of the corresponding limit theorems. As a result, we formulate conditions of convergence, analogous to the classical theory. Limit theorems for probability distributions of products of asymptotically uniformly small random elements on Lie groups are proved as an application of the corresponding statements for manifolds. We assume no conditions related to the commutative property of measure convolutions. [Truncated to < 400 words by viXra Admin]
Category: General Mathematics
[6] viXra:2110.0121 [pdf] submitted on 2021-10-21 14:35:36
Authors: Abhishek Majhi, Ramkumar Radhakrishnan
Comments: 14 Pages.
Given “ab = 0”, considering the arithmetic truth “0.0 = 0” we conclude that one possibility is “both a = 0 and b = 0”. Consequently, the roots of a quadratic equation are mutually inclusive. Therefore, the concerned variable can acquire multiple identities in the same process of reasoning or, at the same time. The law of identity gets violated, which we call the problem of identity. In current practice such a step
of reasoning is ignored by choice, resulting in the subsequent denial of “0.0 = 0”. Here, we deal with the problem of identity without making such a choice of ignorance. We demonstrate that the concept “identity of a variable” is meaningful only in a given context and does not have any significance in isolation other than the symbol, that symbolizes the variable, itself. We demonstrate visually how we actually realize multiple identities of a variable at the same time, in practice, in the context of a given quadratic equation. In this work we lay the foundations, based on which we intend to bring forth some hitherto unattended facets of
reasoning that concern two basic differential equations which are pivotal to the literature of physics.
Category: General Mathematics
[5] viXra:2110.0116 [pdf] replaced on 2022-01-23 18:32:44
Authors: Baoyuan Duan
Comments: 9 Pages.
Build a special odd tree model according with (*3+1)/2^k algorithm, to depart odd numbers in different groups.Carry out research into the tree,found that counts of elements in the tree reduce and converge downward one by one layer,values of elements converge downward to 1,and all odd numbers can appear in the tree only one time.Then prove the “3x+1”guess indirectly.At last,find a logic error of deduction of the tree,and prove 3x+1 guess in another ways.In the last section of last version of this paper,build a special identical equation,use its calculation characters prove and search for solution of any odd converge to 1 equation through (*3+1)/2^k operation,and give a solution for this equation,which is exactly same with calculating directly.And give a specific example to verify it,indicate that we can estimate the value of convergence steps n during some middle procedures.Thus prove 3x+1 guess--Collatz Conjecture strictly. Some supplements:The last section is not very detailed due to time,i think omitting part is not hard and critical.we can fully use characters of odd multifying 3,and no more +1 interference again.To build model for estimating steps n,we can use characters of formula (3) is bigger than corresponding part in formula (2),but should avoid trap:when converge to 1,steps can still increase forever.Add convergence condition:if highest bit of t(i) is 2^k,k should be odd,then all steps n>=1 can satisfy with the equation,i say n>=4 in paper is just want enough parts in formula (2) can appear. Some more supplements:watch t(i),its odd part add corresponding odd produced by odd x calculate directly in each step through (*3+1)/2^k should exactly be 2^k;watch 0 bits in odd part in t(i),because of characters of odd multifying 3,it should shift right or bit-count reduce in each step,and its weight in total t(i) should reduce step by step till to 0,when odd part converge to 1...1.Build weight model:value of all 0 bits in odd part/2^(2k),which 2^(2k) is corresponding part in each step,then model value should reduce step by step,could not exist loop!and model value can and must converge to 0,because obviously there is not possible to exist a convegence value,which its corresponding odd part in t(i) is not 1...1,and its model value can remain unchanged in next steps through multify 3 operation.Then odd part must converge to 1...1,could not diverge or converge to another odd.
Category: General Mathematics
[4] viXra:2110.0080 [pdf] submitted on 2021-10-16 09:09:18
Authors: Timothy W. Jones
Comments: 14 Pages.
Using programs written in TI-Basic increasingly challenging quadratics are factored.
Category: General Mathematics
[3] viXra:2110.0058 [pdf] submitted on 2021-10-13 19:15:44
Authors: Clarence Gipbsin, Lamarr Widmer
Comments: 6 Pages.
We present a method which modifies a magic square of even order n and then adds two outer rows and two outer columns to produce a magic square of order n + 2 . The modification of the original square will change half of its numbers and will preserve the equality of sums of the rows, columns, and main diagonals. This modified square will be centrally embedded in the magic square of order n + 2 .
Category: General Mathematics
[2] viXra:2110.0046 [pdf] submitted on 2021-10-10 02:36:10
Authors: Yuji Masuda
Comments: 1 Page.
Length √x , that the x is a prime number , form the center O to the another circle center can be set in equilateral triangle-square model composed of circles with diameter a. There are various possible values of x depending on a. Among them, x must have some special structure to be a prime number. One possibility is penta-graphene.
Category: General Mathematics
[1] viXra:2110.0014 [pdf] submitted on 2021-10-05 14:43:48
Authors: Viola Maria Grazia
Comments: 1 Page.
In this paper we see the complexes in other point of view
Category: General Mathematics
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