Three tungsten Langmuir probes retrieved from the JET tokamak with the ITER-Like Wall (JET-ILW) from the bulk tungsten Tile 5 have been studied. Nano-indentation, microscopy, ion beam analysis (IBA), and X-ray diffraction were used to assess changes in their mechanical properties, microstructure, and phase composition. Four regions of the probes were studied - the tip and the base, at two sides: front and back. The hardness value of one of the probes (no. 5, Stack B) in the tip area was reduced when compared to the value measured on the base section: 5.4 GPa versus 8.8 GPa, respectively. On the two other probes, the hardness was similar to that of the reference material. At the protrusion of probe 5, the recrystallized zone was observed. The IBA analysis revealed that the probes’ surfaces below the tips were covered by a thin layer of deposit composed primarily of beryllium, oxygen, carbon, and hydrogen isotopes, along with smaller amounts of nickel, nitrogen, and helium at some locations. The presence of tungsten carbide W2C was revealed on the tip of probe 5, in the area where IBA measurements indicated elevated carbon content in the material, demonstrated by analysis of the XRD records.

We show that along a density one subsequence of admissible radii, the nearest neighbor spacing between lattice points on circles is Poissonian.

We investigate function field analogs of the distribution of primes, and prime k-tuples, in “very short intervals” of the form I(f):= {f(x) + a: a ∈ Fp } for f(x) ∈ Fp [x] and p prime, as well as cancellation in sums of function field analogs of the Möbius µ function and its correlations (similar to sums appearing in Chowla’s conjecture). For generic f, i.e., for f a Morse polynomial, the error terms are roughly of size O(√p) (with typical main terms of order p). For non-generic f we prove that independence still holds for “generic” set of shifts. We can also exhibit examples for which there is no cancellation at all in Möbius/Chowla type sums (in fact, it turns out that (square root) cancellation in Möbius sums is equivalent to (square root) cancellation in Chowla type sums), as well as intervals where the heuristic “primes are independent” fails badly. The results are deduced from a general theorem on correlations of arithmetic class functions; these include characteristic functions on primes, the Möbius µ function, and divisor functions (e.g., function field analogs of the Titchmarsh divisor problem can be treated). We also prove analogous, but slightly weaker, results in the more delicate fixed characteristic setting, i.e., for f(x) ∈ Fq [x] and intervals of the form f(x) + a for a ∈ Fq, where p is fixed and q = pl grows.

The Seba billiard, a rectangular torus with a point scatterer, is a popular model to study the transition between integrability and chaos in quantum systems. Whereas such billiards are classically essentially integrable, they may display features such as quantum ergodicity [11], which are usually associated with quantum systems whose classical dynamics is chaotic. Seba proposed that the eigenfunctions of toral point scatterers should also satisfy Berry's random wave conjecture, which implies that the value distribution of the eigenfunctions ought to be Gaussian. However, Keating, Marklof, and Winn formulated a conjecture that suggested that Seba billiards with irrational aspect ratio violate the random wave conjecture, and we show that this is indeed the case. More precisely, for tori having diophantine aspect ratio, we construct a subsequence of the set of new eigenfunctions having even/even symmetry, of essentially full density, and show that its 4th moment is not consistent with a Gaussian value distribution. In fact, given any set Lambda interlacing with the set of unperturbed eigenvalues, we show non-Gaussian value distribution of the Green's functions G(lambda), for lambda in an essentially full density subsequence of Lambda.

We consider momentum push-forwards of measures arising as quantum limits (semiclassical measures) of eigenfunctions of a point scatterer on the standard flat torus T-2 = R-2/Z(2). Given any probability measure arising by placing delta masses, with equal weights, on Z(2)-lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and are completely delocalized in position (i.e., concentration on Lagrangian states). We also show that the mass, in momentum, can fully localize on more exotic measures, for example, singular continuous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals, enabling precise control of the location of perturbed eigenvalues.

We show that arithmetic toral point scatterers in dimension three (“Šeba billiards on R3/ Z3”) exhibit strong level repulsion between the set of “new” eigenvalues. More precisely, let Λ : = { λ1< λ2< … } denote the unfolded set of new eigenvalues. Then, given any γ> 0 , |{i≤N:λi+1-λi≤ϵ}|N=Oγ(ϵ4-γ)as N→ ∞ (and ϵ> 0 small.) To the best of our knowledge, this is the first mathematically rigorous demonstration of a level repulsion phenomena for the quantization of a deterministic system.

We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane H. The angles of lattice points arising from the orbit of the modular group PSL2(Z), and lying on hyperbolic circles, are shown to be equidistributed for generic radii. However, the angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of Z2-lattice points (with certain parity conditions) lying on circles in R2, along a thin subsequence of radii. A notable difference is that measures in the hyperbolic setting can break symmetry; on very thin subsequences they are not invariant under rotation by π/2, unlike in the Euclidean setting where all measures have this invariance property.

We study some counting questions concerning products of positive integers u1, . . . , un which form a nonzero perfect square, or more generally, a perfect k -th power. We obtain an asymptotic formula for the number of such integers of bounded size and in particular improve and generalize a result of D. I. Tolev (2011). We also use similar ideas to count the discriminants of number fields which are multiquadratic extensions of Q and improve and generalize a result of N. Rome (2017)

We prove a Polya-Vinogradov type variation of the Chebotarev density theorem for function fields over finite fields valid for "incomplete intervals" I subset of F-p, provided (p(1/2) log p)/|I| = o(1). Applications include density results for irreducible trinomials in F-p[x], i.e. the number of irreducible polynomials in the set {f(x) = x(d) + a(1)x + a(0) is an element of F-p[x]}a(0) is an element of I-0,I- a(1) is an element of I-1 is similar to |I-0|.|I-1|/d provided |I-0| > p(1/2+is an element of), |I-1| > p(is an element of), or |I-1| > p(1/2+is an element of), |I-0| > p

We study the defect (or 'signed area') distribution of standard toral Laplace eigenfunctions restricted to shrinking balls of radius above the Planck scale, either for deterministic eigenfunctions averaged w.r.t. the spatial variable, or in a random Gaussian scenario ('arithmetic random waves'). In either case we exploit the associated symmetry of the eigenfunctions to show that the expectation (spatial or Gaussian) vanishes. In the deterministic setting, we prove that the variance of the defect of flat eigenfunctions, restricted to balls shrinking above the Planck scale, vanishes for 'most' energies. Hence the defect of eigenfunctions restricted to most of the said balls is small. We also construct 'esoteric' eigenfunctions with large defect variance, by choosing our eigenfunctions so that to mimic the situation on the hexagonal torus, thus breaking the symmetries associated to the standard torus. In the random Gaussian setting, we establish various upper and lower bounds for the defect variance w.r.t. the Gaussian probability measure. A crucial ingredient in the proof of the lower bound is the use of Schmidt's subspace theorem.