Elliptic integral evaluations of Bessel moments

We record what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. More generally, we develop formulae for integrals of products of six or fewer Bessel functions. In consequence, we are able to discover and prove closed forms for $c_{n,k}:=\int_0^\infty t^k K_0^n(t) {\rm d}t$ with integers $n=1,2,3,4$ and $k\ge0$, obtaining new results for the even moments $c_{3,2k}$ and $c_{4,2k}$. We also derive new closed forms for the odd moments $s_{n,2k+1}:=\int_0^\infty t^{2k+1}I_0^{}(t) K_0^{n-1}(t) {\rm d}t$ with $n=3,4$ and for $t_{n,2k+1}:=\int_0^\infty t^{2k+1}I_0^2(t) K_0^{n-2}(t) {\rm d}t$ with $n=5$, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of $s_{5,2k+1}$, make substantial progress on the evaluation of $c_{5,2k+1}$, $s_{6,2k+1}$ and $t_{6,2k+1}$ and report more limited progress regarding $c_{5,2k}$, $c_{6,2k+1}$ and $c_{6,2k}$. In the process, we obtain 8 conjectural evaluations, each of which has been checked to 1200 decimal places. One of these lies deep in 4- dimensional quantum field theory and two are probably provable by delicate combinatorics. There remains a hard core of five conjectures whose proofs would be most instructive, to mathematicians and physicists alike.Comment: 51 pages, 1 Postscript figure, uses amsmath.sty, added reference

Identifying anisotropy in seemingly random CNT networks using terahertz techniques

Characterizing and understanding carbon nanotube (CNT)-based polymer composites plays a crucial role for the ability to customize their properties. The (an)isotropy of CNT networks inside a composite can have significant effects on the final material's properties. However, characterizing the (an)isotropy of seemingly random CNT networks can be difficult using standard techniques. In this letter, we show that terahertz polarization sensitive measurements can provide a reliable, noninvasive and fast way of identifying anisotropy in seemingly homogenous CNT networks as based on criteria by standard techniques

Interinstitutional dimension concerning planning, training and force engagement as response to the hybrid war

The proliferation of risks and unconventional threats, especially the hybrid ones, requires the finding of integrated security solutions, both nationally and internationally. The beginning of the millennium reveals new ideas for conducting military conflicts. Thus, within the future confrontations characterized by a high degree of complexity, awareness of the need and development of some mechanisms necessary for the inter-institutional integration and the effects of the actions of all power tools, military and civilian, is a priority of major significance. In this regard, the present article presents some mechanisms, guidelines and methods that could lead to inter-institutional integration

INTERINSTITUTIONAL DIMENSION CONCERNING PLANNING, TRAINING AND FORCE ENGAGEMENT AS RESPONSE TO THE HYBRID WAR

The proliferation of risks and unconventional threats, especially the hybrid ones, requires the finding of integrated security solutions, both nationally and internationally. The beginning of the millennium reveals new ideas for conducting military conflicts. Thus, within the future confrontations characterized by a high degree of complexity, awareness of the need and development of some mechanisms necessary for the inter-institutional integration and the effects of the actions of all power tools, military and civilian, is a priority of major significance. In this regard, the present article presents some mechanisms, guidelines and methods that could lead to inter-institutional integration

Quantitative conductive atomic force microscopy on single-walled carbon nanotube-based polymer composites

Conductive atomic force microscopy (C-AFM) is a valuable technique for correlating the electrical properties of a material with its topographic features and for identifying and characterizing conductive pathways in polymer composites. However, aspects such as compatibility between tip material and sample, contact force and area between the tip and the sample, tip degradation and environmental conditions render quantifying the results quite challenging. This study aims at finding the suitable conditions for C-AFM to generate reliable, reproducible, and quantitative current maps that can be used to calculate the resistance in each point of a single-walled carbon nanotube (SWCNT) network, nonimpregnated as well as impregnated with a polymer. The results obtained emphasize the technique's limitation at the macroscale as the resistance of these highly conductive samples cannot be distinguished from the tip-sample contact resistance. Quantitative C-AFM measurements on thin composite sections of 150-350 nm enable the separation of sample and tip-sample contact resistance, but also indicate that these sections are not representative for the overall SWCNT network. Nevertheless, the technique was successfully used to characterize the local electrical properties of the composite material, such as sample homogeneity and resistance range of individual SWCNT clusters, at the nano- and microscale

Tip-enhanced Raman mapping of single-walled carbon nanotube networks in conductive composite materials

Identifying and characterizing the structural integrity of single-walled carbon nanotubes (SWCNTs) that are fully embedded in a polymer matrix without causing any damage to them is a difficult task to achieve for bulk samples. Using tip-enhanced Raman spectroscopy, the surface of a polymer-embedded conductive network of SWCNTs was mapped underneath a thin layer of pure polymer. The technique was also used to detect tube-breaking within the composite sample caused by mechanical stress, beyond the 'visual' capabilities of scanning electron microscopy techniques. Results show that tip-enhanced Raman mapping can be used to successfully identify and characterize SWCNTs even underneath a layer of polymer

Single-Walled Carbon Nanotube Networks: The Influence of Individual Tube–Tube Contacts on the Large-Scale Conductivity of Polymer Composites

Over two decades after carbon nanotubes started to attract interest for their seemingly huge prospects, their electrical properties are far from being used to the maximum potential. Composite materials based on carbon nanotubes still have conductivities several orders of magnitude below those of the tubes themselves. This study aims at understanding the reason for these limitations and the possibilities to overcome them. Based on and validated by real single-walled carbon nanotube (SWCNT) networks, a simple model is developed, which can bridge the gap between macroscale and nanoscale down to individual tube–tube contacts. The model is used to calculate the electrical properties of the SWCNT networks, both as-prepared and impregnated with an epoxy-amine polymer. The experimental results show that the polymer has a small effect on the large-scale network resistance. From the model results it is concluded that the main contribution to the conductivity of the network results from direct contacts, and that in their presence tunneling contacts contribute insignificantly to the conductivity. Preparing highly conductive polymer composites is only possible if the number of direct, low-resistance contacts in the network is sufficiently large and therefore these direct contacts play the key role