Papers by Daniele Lattanzi
Journal of advances in mathematics and computer science, Jan 31, 2024
A heuristic i.e. empirical approach to the problem of prime number gaps of many kinds and types, ... more A heuristic i.e. empirical approach to the problem of prime number gaps of many kinds and types, different degrees and orders, treated as simple raw experimental data from the statistical viewpoint is presented. The aim of the article is to show a picture of the actual situation of prime number gaps in order to describe and to try to understand the structure itself of prime gaps of various kinds and orders as well as of primes themselves. The data base comprises the finite sequences of prime number gaps up to the value Pn of the prime counter n = 5•10 7 that is P5E7 = P(5•10 7) = 982,451,653 all of them available in the net. The statistical distributions of prime gaps are best-fitted by the pseudo-Voigt fit function, a convolution of the Lorentz and the Gauss differential distribution functions, or by the so-called E-exp or exp-exp differential distribution function or by a log-linear histogram according to the kind of gaps examined, either δ i Pn (higher order gaps) or Δ k Pm = Pm-Pm-k (deltalags) with i and k ≥ 2 or the simple linear differences δ 1 Pm = Δ 1 Pm = ΔPm= Pm-Pm-1 respectively. One of the unexpected results of the investigation is the appearance of inner structures at high values of nΔ, the number of the intervals of the distributions, suggesting the presence of groups or clusters strictly linked to the nature of prime numbers themselves in which the same phenomenology is present.
British Journal of Mathematics & Computer Science, Jan 10, 2016
The present article shows a proof of Riemann's Hypothesis (RH) which is both general (i.e. valid ... more The present article shows a proof of Riemann's Hypothesis (RH) which is both general (i.e. valid for all the non-trivial zeroes of the zeta function) and elementary (that is not using the theory of the complex functions) in which the real constant σ=+1/2 arises by itself and automatically. The modified chi-square function in one of its four forms (±1/•)X k 2 (Ω,x/ω) is used as an interpolating function of the progressions {n ±α }, of their summations {∑n ±α } and of the progressions {N ±α+1 /(±α+1)}, with α∈R n,N∈N so that k=2±2α and in the real plane (α,k) two half-lines are set up with k<2. The use of the Euler-MacLaurin formula with the oneto-one correspondence between the summation operation ∑ and the shift vector operator Σ≡(Σ α ,Σ k) in the real 2D plane (α,k) lead to find the zeroes of Euler's function. Finally, the extrusion to the third imaginary axis it leads to prove Riemann's hypothesis.
Scale Laws of Prime Number Frequencies by the Modified Chi-square Function
British Journal of Mathematics & Computer Science, Jan 10, 2016
British Journal of Mathematics & Computer Science, Jan 10, 2017
An innovative approach that treats prime numbers as raw experimental data and as elements of larg... more An innovative approach that treats prime numbers as raw experimental data and as elements of larger and larger finite sequences {P m }≡{P(m p)} is shown in the present report. The modified chi-square function X k 2 (A,m p /x o) with its three parameters A, k and x o =x o (k) is the best-fit function of the finite sequences {ρ m }≡{lgP m /lgm p } from the analytical viewpoint thus showing that the property of scale invariance does not hold for the finite sequences of this prime variable and so for primes themselves. In addition an injective map can be set between these {ρ m } sequences and the {m α } progressions with domain N and codomain R + being α∈(-1,0)⊂Rthrough the parameter k=2+2α of their common fit function X k 2 (A,m p /x o). All that leads to induction algorithms and to relationships of the kind P m ≈P(m p), though within the precisions of the calculations and holding locally.
Journal of Advances in Mathematics and Computer Science
Prime numbers represent one of the major open problems in number theory mostly in that at present... more Prime numbers represent one of the major open problems in number theory mostly in that at present it is not possible to state that the induction principle holds for them. The methodology of experimental mathematics has been little endeavored in this field thus the present report deals with an innovative approach to the problem of primes treated as raw experimental data and as elements of larger and larger finite sequences {Pn}. The modified Chi-square function in the form -1/X2k(A,n/μ) with the ad-hoc A, k and μ parameters is the best-fit function of the finite sequences of primes {Pn}, like the truncated progressions {Cαnα} with domain N and co-domain R+, being (α,k)≡(1+,0-) and k=2-2α and just like the function λn×n×ln(n), what leads to induction algorithms and to many fit relationships Pn≈P(n) though within the precisions of the calculations that is approximate. A bi-injective map can be set between the prime sequences and any of these three fit functions showing that the propert...
The statistical distribution of prime numbers represents an open problem in number theory still n... more The statistical distribution of prime numbers represents an open problem in number theory still nowadays. The methodology of experimental mathematics has not yet been attempted in this field, thus the present report treats prime numbers as raw experimental data and as elements of larger and larger finite sequences {Pm}. The modified chi-square function X k(A, x/μ) with the ad-hoc A, k and μ = μ(k) parameters is the best-fit function of the differential distribution functions of both prime finite sequences {Pm} and truncated progressions {n} with α ∈ (1, 2) so that an injective map can be set between them through the parameter k of their common fit function X k(A, x/μ) showing that the property of scale invariance does not hold for prime distribution. The histograms of prime gaps, which are best fitted by standard statistical distribution functions, show unexpected clustering effects.
British Journal of Mathematics & Computer Science, 2017
An innovative approach that treats prime numbers as raw experimental data and as elements of larg... more An innovative approach that treats prime numbers as raw experimental data and as elements of larger and larger finite sequences {P m }≡{P(m p)} is shown in the present report. The modified chi-square function X k 2 (A,m p /x o) with its three parameters A, k and x o =x o (k) is the best-fit function of the finite sequences {ρ m }≡{lgP m /lgm p } from the analytical viewpoint thus showing that the property of scale invariance does not hold for the finite sequences of this prime variable and so for primes themselves. In addition an injective map can be set between these {ρ m } sequences and the {m α } progressions with domain N and codomain R + being α∈(-1,0)⊂Rthrough the parameter k=2+2α of their common fit function X k 2 (A,m p /x o). All that leads to induction algorithms and to relationships of the kind P m ≈P(m p), though within the precisions of the calculations and holding locally.
British Journal of Mathematics & Computer Science, 2016
The present article shows a proof of Riemann's Hypothesis (RH) which is both general (i.e. valid ... more The present article shows a proof of Riemann's Hypothesis (RH) which is both general (i.e. valid for all the non-trivial zeroes of the zeta function) and elementary (that is not using the theory of the complex functions) in which the real constant σ=+1/2 arises by itself and automatically. The modified chi-square function in one of its four forms (±1/•)X k 2 (Ω,x/ω) is used as an interpolating function of the progressions {n ±α }, of their summations {∑n ±α } and of the progressions {N ±α+1 /(±α+1)}, with α∈R n,N∈N so that k=2±2α and in the real plane (α,k) two half-lines are set up with k<2. The use of the Euler-MacLaurin formula with the oneto-one correspondence between the summation operation ∑ and the shift vector operator Σ≡(Σ α ,Σ k) in the real 2D plane (α,k) lead to find the zeroes of Euler's function. Finally, the extrusion to the third imaginary axis it leads to prove Riemann's hypothesis.
21st IEEE/NPS Symposium on Fusion Engineering SOFE 05, 2005
Polycrystalline chemical vapor deposited (CVD) diamond detectors are of great interest in harsh e... more Polycrystalline chemical vapor deposited (CVD) diamond detectors are of great interest in harsh environments due to their capability to operate even at high temperature and to withstand great neutron fluencies. Their use is of interest in many fields including the nuclear fusion reactors. In this paper the successful characterization of polycrystalline CVD diamond detectors as 14 MeV neutron monitors is reported. Two polycrystalline CVD diamond detectors of 96 µm and 126 µm thickness respectively were used. The detectors' count rate versus neutron flux was studied for various applied electric fields, showing that these detectors have a linear behavior in the flux range explored. The detector efficiency was also derived resulting in stable behavior vs. the neutron flux. The low sensitivity to gamma-ray field and the possibility to reject the counts due to gamma-rays was also assessed. The time dependent neutron emission from the 14 MeV Frascati Neutron Generator (FNG) was recorded with CVD diamond detectors. The data for diamonds were compared with that recorded by the standard monitors available at FNG showing good agreement. Good stability and capability to operate (with negligible pileup effect) with neutron flux up to 3x10 8 n cm-2 s-1 was observed. Following this characterization one diamond detector was installed at JET and successfully operated during the trace tritium experiment campaign as a flux monitor of the 14 MeV neutron emission from JET.
Detailed electromagnetic analyses of the ITER in-vessel components during plasma disruptions
17th IEEE/NPSS Symposium Fusion Engineering (Cat. No.97CH36131), 1998
The electromagnetic (EM) loads during plasma disruptions represent one of the main concern for th... more The electromagnetic (EM) loads during plasma disruptions represent one of the main concern for the ITER in-vessel components. To reduce their effects on the mechanical structures some blanket and divertor design changes have been recently introduced. The effectiveness of these changes has been checked for some of the most critical disruption events, centred disruptions and vertical displacement event disruptions. In order to evaluate the contribution to the EM loads of the halo currents as well as of the currents induced by the thermal quench, a plasma model has also been developed
Feasibility study of a toroidal limiter for the FTU machine
[Proceedings] The 14th IEEE/NPSS Symposium Fusion Engineering
The Frascati Tokamak Upgrade (FTU) is now operating in the ohmic phase (Ip up to 1 MA Bt=6 T) wit... more The Frascati Tokamak Upgrade (FTU) is now operating in the ohmic phase (Ip up to 1 MA Bt=6 T) with a poloidal limiter. The large heat flux expected during the 8-MW, 8-GHz LH radio frequency heating experimental phase suggested the design and verification of the feasibility of a toroidal limiter. The analyses were carried out on both stainless steel and graphite. The results obtained demonstrate that the solution providing graphite tiles is much better in terms of mechanical performance and reliability
Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2008
During the 2005-2007 JET experimental campaigns a Single Crystal Diamond (SCD) detector covered w... more During the 2005-2007 JET experimental campaigns a Single Crystal Diamond (SCD) detector covered with a 2-mm thick LiF film 95% enriched in 6 Li was installed at JET and operated continuously during the whole experimental campaign with the goal to measure both the total and the time-dependent neutron emission from plasmas. After reviewing the history of diamond as neutron detector for tokamak application, this paper reports on the experimental measurements performed during that time window. The results are compared with those obtained with detectors routinely working at JET (Fission Chambers and silicon diodes). In particular it was demonstrated that using an SCD detector covered with a 6 LiF film it has been possible to discriminate between the total and the 14-MeV neutrons coming from triton burn-up. This allowed, for the first time using a single detector, the simultaneous measurement of both the total and 14-MeV neutron yields as well as their time dependency with very good reliability and stability.
IEEE Transactions on Nuclear Science, 2009
This paper reports about state-of-the-art artificial Single Crystal Diamond (SCD) neutron detecto... more This paper reports about state-of-the-art artificial Single Crystal Diamond (SCD) neutron detectors based on a multilayered structure and grown by chemical vapour deposition (CVD) technique. Multilayered SCD detectors covered with a thin layer of 6 LiF allow the simultaneous detection of both slow and fast neutrons and can operate in pulse and current mode. These detectors can also be produced with a thin layer of Boron. Application of SCD detectors to neutron detection around fusion tokamak is reported. Some problems related to the processing of the very fast electrical pulse produced by diamond are addressed and the achieved and foreseen development of the processing electronics is reported as well.
Fusion Engineering and Design, 2007
Artificial diamond neutron detectors recently proved to be interesting devices to measure the neu... more Artificial diamond neutron detectors recently proved to be interesting devices to measure the neutron production on large experimental fusion machines. Diamond detectors are very promising detectors to be used in fusion environment due to their radiation hardness, low sensitivity to gamma rays, fast response and high energy resolution. High quality "electronic grade" diamond films are produced through microwave Chemical Vapour Deposition (CVD) technique. Two CVD diamond detectors have been installed and operated at Joint European Torus (JET), Culham Science Centre, UK. One of these detectors was a polycrystalline CVD diamond film, about 12mm 2 area and 30µm thickness while the second was a monocrystalline film of about 5mm 2 area and 20µm thick. Both diamonds were covered with 2µm of Lithium Fluoride (LiF) 95% enriched in 6 Li. The LiF layer works as a neutron-to-charged particle converter so these detectors can measure thermalized neutrons. Their output signals were compared to JET total neutron yield monitors (KN1 diagnostic) realized with a set of Uranium fission chambers. Despite their small active volumes theour diamond detectors were able to measure total neutron yields with good reliability and stability during the recent JET experimental campaign of 2006.
Fusion Engineering and Design, 2009
This paper reports on the three new Single Crystal Diamond (SCD) detectors installed at JET for t... more This paper reports on the three new Single Crystal Diamond (SCD) detectors installed at JET for the 2008 campaigns. The yields of both total and 14-MeV neutrons produced during DD plasma pulses as well as the time dependent neutron emission have been measured. The first detector, installed in the Vertical Port (Oct-1) of JET, is 200µm thick and is embedded in paraffin. It has a detection efficiency of about 2.9*E-05 counts/n * cm 2 for the 14MeV neutrons. The second detector, located in Oct-1 Limb 1 / 2 , is 104µm thick and is covered by a thermally evaporated 3 ºm thick 6 LiF film in order to detect the total and the 14MeV neutrons. In addition, it is surrounded by a 2.5-cm-thick polyethylene shield to enhance its thermal neutron response. The third detector is mounted in the main horizontal port (Oct-6) and it is operated in an innovative way, that is with a single low capacitance super screened cable and the whole electronic chain is outside the JET Torus Hall. Furthermore it uses fast electronics, suitable to the fast diamond response pulse (<1ns). It is 75µm thick, covered by a 3µm thick 6 LiF film and surrounded by a 2.5cm-thick layer of polyethylene. All the detectors have been previously tested and qualified at the ENEA-Frascati Neutron Generator with 2.5MeV and 14.7MeV neutron beams. After the description of their main features, the results of the measurements so far performed are reported showing a very good matching with other standard detectors, such as fission chambers and Si diodes, routinely used at JET.
Neutron Spectroscopy by Means of Artificial Diamond Detectors Using a Remote Read Out Scheme
IEEE Transactions on Nuclear Science, 2010
Artificial crystal diamond neutron detectors are under test at JET tokamak since 2003 and they ha... more Artificial crystal diamond neutron detectors are under test at JET tokamak since 2003 and they have demonstrated to be reliable and stable as well as to withstand the harsh working condition available in a large tokamak. Up to now they were used to measure the total and time dependent neutron emission while neutron spectroscopy was never attempted. On the other
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Papers by Daniele Lattanzi