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Jorma Jormakka

Publications and source records attributed to Jorma Jormakka.

9 recordsLinked to original sources
arXiv2021

Calculating Cost Distributions of a Multiservice Loss System

Congestion pricing has received lots of attention in scientific discussion. Congestion pricing means that the operator increases prices at the time of congestion and the traffic demand are expected to decrease. In a certain sense, shadow prices are an optimal way of congestion pricing: users are charged shadow prices, i.e., the expectations of future losses because of blocked connections. The shadow prices can be calculated exactly from the Howard equation, but this method is difficult. The paper presents simple approximations to the solution of the Howard equation and a way to derive more exact approximations. If users do not react by lowering their demand, they will receive higher bills to pay. Many users do not react to increased prices but would want to know how the congestion pricing mechanism affects the bills. The distribution of the price of a connection follows from knowing the shadow prices and the probability of a congestion state. There is another interesting distribution. The network produces profit to the operator, or equivalently, blocked connections produce a cost to the operator. The average cost rate can be calculated from the Howard equation, but the costs have some distribution. The distribution gives the risk that the actual costs exceed the average costs, and the operator should include this risk in the prices. The main result of this paper shows how to calculate the distribution of the costs in the future for congestion pricing by shadow prices and for congestion pricing with a more simple pricing scheme that produces the same average costs.

math.OC↗
arXiv2021

A Theorem of Congruent Primes

To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter than existing proofs, but the method may be useful in other contexts. The proof of Theorem 1 tracks the set of solutions and this set branches as a binary tree. Conditions set to the theorem restricts the branches so that only one branch is left. Following this branch gives either a solution or a contradiction. In Theorem 1 it leads to a contradiction. The interest is in the proof method, which maybe can be generalized to non-primes.

math.NT↗
arXiv2021

On the zeros of the Riemann zeta function

In 2008 I thought I found a proof of the Riemann Hypothesis, but there was an error. In the Spring 2020 I believed to have fixed the error, but it cannot be fixed. I describe here where the error was. It took me several days to find the error in a careful checking before a possible submission to a payable review offered by one leading journal. There were three simple lemmas and one simple theorem, all were correct, yet there was an error: what Lemma 2 proved was not exactly what Lemma 3 needed. So, it was the connection of the lemmas. This paper came out empty, but I have found a different proof of the Riemann Hypothesis and it seems so far correct. In the discussion at the end of this paper I raise a matter that I think is of importance to the review process in mathematics.

math.GM↗
arXiv2020

On the existence of polynomial-time algorithms to the subset sum problem

This paper proves that there does not exist a polynomial-time algorithm to the the subset sum problem. As this problem is in NP, the result implies that the class P of problems admitting polynomial-time algorithms does not equal the class NP of problems admitting nondeterministic polynomial-time algorithms.

math.GM↗
arXiv2012

Solutions to 3-dimensional Navier-Stokes equations for incompressible fluid

This article is an updated version of the article that was published in the Electronic Journal of Differential Equations on 10. July 2010. Two footnotes have been added. One corrects a minor error not influencing the proof, the second is only a clarifying text to the existing proof. A discussion how the published article solves the Clay Millennium Prize problem on the Navier-Stokes equations is added, the critizism against the published article is answered and a discussion how the Clay problem statement should be corrected is included. The article gives explicit solutions to the space-periodic Navier-Stokes problem with non-periodic pressure. These type of solutions are not unique and by using such solutions one can construct a periodic, smooth, divergence-free initial vector field allowing a space-periodic and time-bounded external force such that there exists a smooth solution to the 3-dimensional Navier-Stokes equations for incompressible fluid with those initial conditions, but the solution cannot be continued to the whole space.

math.GM↗
arXiv2010

Solutions to Yang-Mills equations

This article gives explicit solutions to the Yang-Mills equations. The solutions have positive energy that can be made arbitrarily small by selection of a parameter showing that Yang-Mills field theories do not have a mass gap.

math.GM↗