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Register a new userDid Hypatia Know about Negative Numbers?
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Added by
Marek A. Abramowicz
Publication date
2012
fields
Physics
and research's language is
English
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In this Letter we comment on one particular aspect of Hypatias enigmatic biography by translating into English a short poem that appeared in a recent review of the third revised Polish edition of Maria Dzielskas book about Hypatia. It poses a simple and specifc question: did Hypatia know about the negative numbers?
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String theory has transformed our understanding of geometry, topology and spacetime. Thus, for this special issue of Foundations of Physics commemorating Forty Years of String Theory, it seems appropriate to step back and ask what we do not understand. As I will discuss, time remains the least understood concept in physical theory. While we have made significant progress in understanding space, our understanding of time has not progressed much beyond the level of a century ago when Einstein introduced the idea of space-time as a combined entity. Thus, I will raise a series of open questions about time, and will review some of the progress that has been made as a roadmap for the future.
Translated from the Latin original Novae demonstrationes circa resolutionem numerorum in quadrata (1774). E445 in the Enestrom index. See Chapter III, section XI of Weils Number theory: an approach through history. Also, a very clear proof of the four squares theorem based on Eulers is Theorem 370 in Hardy and Wright, An introduction to the theory of numbers, fifth ed. It uses Theorem 87 in Hardy and Wright, but otherwise does not assume anything else from their book. I translated most of the paper and checked those details a few months ago, but only finished last few parts now. If anything isnt clear please email me.
Temperature, the central concept of thermal physics, is one of the most frequently employed physical quantities in common practice. Even though the operative methods of the temperature measurement are described in detail in various practical instructions and textbooks, the rigorous treatment of this concept is almost lacking in the current literature. As a result, the answer to a simple question of what the temperature is is by no means trivial and unambiguous. There is especially an appreciable gap between the temperature as introduced in the frame of statistical theory and the only experimentally observable quantity related to this concept, phenomenological temperature. Just the logical and epistemological analysis of the present concept of phenomenological temperature is the kernel of the contribution.
In the present paper, we investigate the cosmographic problem using the bias-variance trade-off. We find that both the z-redshift and the $y=z/(1+z)$-redshift can present a small bias estimation. It means that the cosmography can describe the supernova data more accurately. Minimizing risk, it suggests that cosmography up to the second order is the best approximation. Forecasting the constraint from future measurements, we find that future supernova and redshift drift can significantly improve the constraint, thus having the potential to solve the cosmographic problem. We also exploit the values of cosmography on the deceleration parameter and equation of state of dark energy $w(z)$. We find that supernova cosmography cannot give stable estimations on them. However, much useful information was obtained, such as that the cosmography favors a complicated dark energy with varying $w(z)$, and the derivative $dw/dz<0$ for low redshift. The cosmography is helpful to model the dark energy.
This is an English translation of the Latin original De summa seriei ex numeris primis formatae ${1/3}-{1/5}+{1/7}+{1/11}-{1/13}-{1/17}+{1/19}+{1/23}-{1/29}+{1/31}-$ etc. ubi numeri primi formae $4n-1$ habent signum positivum formae autem $4n+1$ signum negativum (1775). E596 in the Enestrom index. Let $chi$ be the nontrivial character modulo 4. Euler wants to know what $sum_p chi(p)/p$ is, either an exact expression or an approximation. He looks for analogies to the harmonic series and the series of reciprocals of the primes. Another reason he is interested in this is that if this series has a finite value (which is does, the best approximation Euler gets is 0.3349816 in section 27) then there are infinitely many primes congruent to 1 mod 4 and infinitely many primes congruent to 3 mod 4. In section 15 Euler gives the Euler product for the L(chi,1). As a modern mathematical appendix appendix, I have written a proof following Davenport that the series $sum_p frac{chi(p)}{p}$ converges. This involves applications of summation by parts, and uses Chebyshevs estimate for the second Chebyshev function (summing the von Mangoldt function).
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