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A simple and fast data processing algorithm for the resolution enhancement of two dimensional (2 D) nuclear magnetic resonance spectra is proposed. It is based on a modified 2 D Laplacian operator and can be used as a standalone algorithm or embedded within peak picking routines. It can also be combined with other processing techniques such as line...

Dissolution dynamic nuclear polarization allows in vivo studies of metabolic flux using 13 C-hyperpolarized tracers by enhancing signal intensity by up to four orders of magnitude. The T1 for in vivo applications is typically in the range of 10-50 s for the different 13 C-enriched metabolic substrates; the exponential loss of polarization due to va...

Simulated data are very important for the development, testing, and comparisons of algorithms for inverse problems (IP) of all kinds. The many inverse problems encountered in magnetic resonance (MR) are no exception. --- Examples of raw MR data to be evaluated in various ways include FID's, relaxation decays, relaxation rate dependences on magnetic...

Field cycling NMR relaxometry is often used to measure the characterize solid and semi-solid samples which can be at least approximately characterized by a single longitudinal relaxation time T1. The focus in these cases is on the field dependence of the T1, i.e., just the apparent dispersion curve. --- However, many system of great importance both...

This invited tutorial was originally supposed to have four Sections but, due to utter lack of time (40'), only two could be covered. The plan is to develop the topic in its full breadth in a book form. --- Section I: Generalities. A brief review of the very basic concepts regarding magnetic particles, including atomic nuclei. This means the angular...

PcBc is an algorithm designed to automatic correct the phase and baseline of a spectrum. The algorithm optimizes a quality function based on the histogram of the spectrum, and the corrections are applied on both the in-phase and the out-of-phase parts of a spectrum. The automatic feature of the algorithm ensures objectivity and reproducibility of t...

A large and growing collection of mathematical constants and sequences, published and maintained online (this pdf is converted from the online HTML format). Started in 31 March 2008, it gets updated on an irregular basis.

Metabolomic spectra are particularly difficult to properly phase and to correct their baseline roll. This is principally due to the presence of very large number of heavily overlapping peaks which are therefore difficult to phase and, in addition, leave only limited (or no) gaps between groups of peaks where the baseline can be visually or programm...

In vivo spectra (1H, 13C, 31P, 23Na, ...) are known to be particularly difficult to phase and to correct for baseline drifts. The reasons are multiple: apart from the usual ones encountered typical of in-vitro spectroscopy, such as post-pulse receiver transfer-function distortions (dead-time), filter delays, filter roll, an a few others, there are...

In many branches of physics, the time evolution of various quantities measured in systems passing from excited to equilibrium states, while theoretically very complex, can be in practice well approximated by a sum of exponential decays or by a continuous distribution. This applies very often also to the signals observed in NMR relaxometry. In those...

Manually, phase (PC) and baseline (BC) corrections in 1D NMR spectra were always done sequentially one after another. In addition, the Bc was always done only on the “real” part of the spectrum and not on the imaginary part. However, one often notices that there is a mutual interference between the two corrections, causing uncertainty about the bes...

The equilibrium state of nuclear magnetization is quite easy to derive from the spin Hamiltonian and from basic principles of statistical physics and thermodynamics [1]. Somewhat less straightforward is predicting the behavior of nuclear magnetization in imposed magnetic fields, which vary either in magnitude, or in direction, or both. In such non-...

In the field of Nuclear Magnetic Resonance, the capability to distinguish/filter the signal of different part of a sample plays a key role in many applications. For example, in the medical field such possibility allows to improve the image contrast and the identification of the different tissue inside the human body. In the oil and gas field, the c...

Manually, phase (PC) and baseline (BC) corrections in 1D NMR spectra are done sequentially one after another. In addition, the Bc is always done only on the 'real' part of the spectrum where the spectral peaks are much narrower than in the imaginary part and therefore interfere less with the visual assessment of the final quality. However, even whe...

It is almost ten years since I started working with Mestrelab Research (Santiago de Compostela) on some aspects of their Mnova software NMR plug-in. One of the tasks I am handling is the automatic structure verification (ASV) which consists in answering the question "to which degree is this proposed molecular structure compatible with this set of N...

The algorithmic problem of phase correction (PC), and that of baseline correction (BC), of 1D NMR spectra have been both tackled many times over the last half a century, by many authors. There are many algorithms which emulate the manual procedure. Basically, they all ‘fit’ the parameters which describe the phase (ph0, ph1) and the baseline (variou...

This Note explores a family of sequences related through their exponential generating functions (e.g.f.) to smooth solutions of the nonlinear differential equation f’’(z) = af’(z)f(z) in C. The implied recurrence for the complex expansion coefficients of the solutions leads to a family of sequences uniquely labeled by two complex parameters. Exclud...

This Note illustrates how even simple mappings often hide a non-intuitive inner richness. The functions exp(z) and its inverse log(z), for example, despite their apparent ‘uneventful smoothness’, define two denumerable sets of complex constants, namely the fixed points of the exponential mappings exp(z) and -exp(z). We analyze these two sets, inclu...

Quantitative (1)H NMR (qNMR) is a widely applied technique for compound concentration and purity determinations. The NMR spectrum will display signals from all species in the sample, and this is generally a strength of the method. The key spectral determination is the full and accurate determination of one or more signal areas. Accurate peak integr...

Spectroscopic methods - of which nuclear magnetic resonance (NMR) is one of the most vital players - have almost entirely replaced those old “wet chemistry”-based approaches and the art has become a science. The fact that any automatic structure verification (ASV) system measures the degree of compatibility between a proposed molecular structure an...

Spectroscopic methods ‐ of which nuclear magnetic resonance (NMR) is one of the most vital players ‐ have almost entirely replaced those old “wet chemistry”‐based approaches and the art has become a science. The fact that any automatic structure verification (ASV) system measures the degree of compatibility between a proposed molecular structure an...

This brief essay explores the limit mean densities of subsets of natural numbers m such that the pair (m, m\b) is either coprime or not coprime. Here m\b = floor(m/b) denotes the operation which can be also characterized as 'truncation of the last digit in base b' or 'right truncation in base b'. The analysis leads to an interesting rational-valued...

Investigating elementary properties of random selections from a finite, discrete set, an identity popped up which can be cast both as a finite probability distribution and as a decomposition of nn. They are reported here, together with a few notes on the original object of this study.

This educational article shows the versatility of an 1826 identity due to Niels Hendrik Abel by using it to derive a large number of corollaries, not all of which are particularly well-known. The author thinks that a brief review of what one can do with Abel’s identity might be useful.

Cycles of various lengthscomposed of elements belonging to a set Scan be used to define monocyclicpermutations (“round-table seatings”) of subsets of S.Thoseareinteresting by themselves, but the whole contextbecomes stillmore interesting when the elements possessspecificpropertiesor when pairs of elementsof Shave a specific property(implying a part...

This article explores cycles of integers sharing the special property of the ?classical? Gray sequence which make the latter so popular in technical applications, namely the fact that the binary expansions of any two adjacent members differ by exactly one digit. However, there are many cycles having this property: given any even number 2n > 0, it i...

Nuclear magnetic resonance (NMR) spectroscopy is a powerful analytical technique, but its low sensitivity and highly sophisticated, costly, equipment severely constrain more widespread applications. Here we show that a non-resonant planar transceiver microcoil integrated in a microfluidic chip (detection volume 25 nl) can detect different nuclides...

A few years ago Don Blazys caused a ripple of excitation among numerologists by presenting a real number which, applying an iterative computation recipe, produced all prime numbers. The procedure is one of an infinity of possible mappings between subsets of real numbers and subsets of integer sequences. In this particular case, the source set is th...

A novel data-evaluation procedure for the automatic atom to peak or multiplet assignment of (1) H-NMR spectra of small molecules has been developed using a fast and robust expert system. The applicability and reliability of the method are demonstrated by comparison of a manually assigned database of (1) H-NMR spectra with the assignments produced b...

NMR is routinely used to quantitate chemical species (qNMR). The necessary experimental procedures to acquire quantitative data are well known, but relatively little attention has been applied to data processing and analysis. We describe here a robust expert system that can be used to automatically choose the best signals in a sample for overall co...

In vivo 1H magnetic resonance spectroscopy (MRS) enables a non-invasive determination of tissue concentrations of various metabolites and compounds in animals or humans. Thus, metabolism can be investigated in vivo, and pathological as well as drug- or exercise-induced changes of the metabolism can be observed. MRS is increasingly being recognized...

The community of NMR spectroscopists certainly does not need an explanation of the meaning of terms like 'structure verification' or 'structure elucidation' or, more generally, 'interpretation of a set of NMR spectra'. Likewise, the addition of terms like 'computer aided' or 'automatic' of the previous terms is quite straightforward and intuitive....

The invention relates to a Nuclear Magnetic Resonance (NMR) spectroscopy device adapted for carrying out 1D and nD homo- and heteronuclear NMR spectroscopy measurements of a plurality of nuclei, comprising an RF coil adapted to transmit RF to and/or receive RF from a measuring volume, wherein the RF coil forms part of a non-tuned radiofrequency cir...

A mildly speculative and humorous reflection for Physicists and Philosophers.

There is a growing need to perform NMR analyses faster, and under full automation control [1]. Advances in available hardware have contributed favourably to this objective, and the routine collection of high-resolution NMR data on large compound libraries is now a relatively simple matter. But without reliable software tools to automatically analys...

For full abstract, please, see the link http://doi.org/10.3247/SL4Nmr12.002 Last para: ... Here we present an integrated application based on an R module called package MUMA (Multivariate & Univariate Metabolomic Analysis) and Mnova software to the processing and analysis of bone tissue biopsies. MUMA was developed by our group and is available onl...

In this work we present an integrated new software solution aimed at the automatic assignment of 1H NMR spectra of small molecules. It is based on an expert system constructed on principles of fuzzy logic with probabilistic methods used for the classification of the different resonances in the spectrum as well as for the enumeration of all the most...

This article presents a series of rational-function approximations for the Dawson integral function F(x) which exhibit correct behavior for very small and very large values of the argument. For any real argument, the worst absolute error of the best approximation presented here is about 2.1e-6, while the worst relative error is 10.0 ppm. These are...

An electric LCR circuit which includes a coil containing magnetic moments can be analyzed using the method of virtual energies which is essentially a thermodynamic paradigm. Such an approach brings forth the primary character of the interaction term between the current and the magnetic moments. It will be shown, for example, that the so-called reci...

GSD is automatically reduces a frequency domain spectrum to a set of Lorentzian or near Lorentzian lines leaving out any baseline drift and noise. A reliable calculation of the derivatives is especially important, particularly in spectra with poor signal-to-noise ratio. Another tricky problem in the case of water is that it does not only show up as...

Parametrically Enabled Relaxation FIlters with Double and multiple Inversion (PERFIDI) is an experimental NMR/MRI technique devised to analyze samples/voxels characterized by multi-exponential longitudinal relaxation. It is based on a linear combination of NMR sequences with suitable preambles composed of inversion pulses. Given any standard NMR/MR...

Objectives: Parametrically Enabled Relaxation Filters with Double and multiple Inversion (PERFIDI) represent a method useful in NMR/MRI to quantify and/or suppress components of NMR signals. They resemble the electronic band-pass, high-pass or low-pass filters, but are used to separate sample components according to their longitudinal relaxation ti...

NMR spectroscopy is an important tool in many metabolomic applications. Its potential capability to handle complex mixtures of metabolites makes it a prime choice in both identification and quantification of the multitude of different species constituting unprocessed biological mixtures. In quantitative metabolomics (targeted metabolomic profiling)...

Bayesian approach to the evaluation of DOSY (and similar) NMR spectra is very fruitful. It provides results of the same quality as MEM (Maximum Entropy Management) but is several orders of magnitude faster! This poster introduces a refinement trick (improved alignment of the diffusing species) to the basic method presented earlier. The algorithm is...

A previous article has introduced the concept of generalized Heronian mean Her(k,a) of an integer rank k for any set a of n non-negative real values. The article proved a number of properties of the new means, but it left open several conjectures, one of which regarded the existence of a limit for large k, Her(a), nicknamed Hermean, which also sati...

A previous article has introduced the concept of generalized Heronian mean Her(k,a) of an integer rank k for any n-tuple a of non-negative real numbers. The article left open several conjectures, one of which being that a generalized Heronian mean of a and of any rank k is never smaller than the geometric mean G(a) of a. In a subsequent note the co...

A previous article has introduced the concept of generalized Heronian mean Her(k,a) of an integer rank k for any set a of n non-negative real values. While the article proved a number of properties of the new means, it left open several conjectures, most notably an general inequality and the existence of a limit for large k, nicknamed Hermean Her(a...

This educational article responds to a query about errors which might be committed when, following the common practice, one neglects scalar relaxation in NOE-based measurements of internuclear distances in proteins. It starts with a very brief, qualitative introduction to the phenomenon of relaxation in MR and follows by an order-of-magnitude asses...

Convex real functions of real variable satisfy many inequalities. The one proved here might be novel, though it is difficult to be sure. In any case, a proof is given and it is hoped that, even if the inequality itself were not new, the proof might contain some innovative aspects. The inequality can be expressed verbally as follows: Let f(x) be a...

The classical concept of the Heronian mean of two non-negative real values is subject to a sweeping generalization in order to make it applicable to any subset a of n elements of R+, the set of all positive real numbers. Moreover, it is also extended to include an integer parameter k called rank. The resulting means, denoted as Her(k,a), coincide...

This educational article can be used as a primer for students and non-mathematicians. I find the otherwise excellent introductions to means in Wikipedia and Math World not quite complete in some respects and not as easily accessible to a layman as they could/should be (in any case, the more such introductory sources there are, the better). Since m...

Everybody knows the meaning of the term "spectral multiplet deconvolution" and most are aware that the algorithms dedicated to this task are by no means simple, since they involve quite extensive-and often somewhat fuzzy-prior knowledge (input other than the experimental data themselves) such as the number of lines in the multiplet, their shapes, a...

This article is part of the Know Thy Spins series. It lists some of the generic characteristics of a coupled AB system composed of two spin 1/2 nuclides and points out the simple geometrical relations between the frequencies, intensities and NMR parameters of its four spectral peaks under standard liquid state conditions (isotropic environment, fa...

In this work we present a newly developed Bayesian approach to the processing of NMR diffusion experiments which is computationally very efficient and physically eminently meaningful. It treats all data points in the same way, and gives very satisfactory and artifact-free results. Applied specifically to DOSY data sets, it leads to what we call the...

A typical experimental NMR spectrum comprises a) spectral peaks arising from the transitions of the studied spin system(s), b) satellite peaks from the very same spin system(s) isotopomers, c) solvent peaks due to the employed solvent, d) impurity peaks due to undesired chemical components of the sample, e) folded-over peaks from outside the spectr...

We have developed a mathematical model to simulate the effects of pulse sequences on composite samples and coded it in Matlab. The object-oriented code accommodates pulse sequences and samples of any complexity, including ones with any distribution of relaxation rates and offsets. Of equal importance is the fact that the most common instrumental ar...

Over the years multidimensional NMR has become an essential tool for the structural analysis of molecules but, in practice, the most widely used experiment is still the 1D proton NMR, which, when properly interpreted, gives a wealth of information with minimal acquisition time and sample quantity. However, a detailed analysis of 1H-NMR, especially...

A well known problem with several types of multi-array NMR techniques is the visual representation of the results in terms of physico-chemical parameters associated with individual spectral peaks. This regards, for example, the Diffusion Ordered Spectroscopy (DOSY) in which the non-spectral parameter is the diffusion coefficient and ROSY (Relaxatio...

We have developed a family of fast algorithms [1] which opens new avenues in NMR data processing and evaluation. This poster illustrates the results obtained with the Resolution BoosterTM - a member of the family which narrows and isolates spectral peaks in 1D and 2D spectra. The Resolution Booster operates on frequency domain data (either complet...

We have developed a family of fast algorithms [1] which opens new avenues in NMR data evaluation. This poster illustrates the results obtained with what we call the J-CorrelatorTM - a member of the family which permits to carry out fast, visual correlation between various spectral-line splittings in complex 1D NMR spectra.

This educational article attempts to explain in simple, non mathematical terms the essence of the Hebel-Slichter and its experimental setup. To cite the article, please use the format: Stan Sykora, The Hebel-Slichter effect, Stan's Library (ISSN 2421-1230), Volume II, February 13, 2008, DOI 10.3247/SL2Nmr08.001.

This educational article is intended for those who have never heard about the antenna reciprocity theorem and its role in Magnetic Resonance. Using a purely phenomenological approach, it explains the essence of the theorem and provides indications about its applications.

Though NMR spectra contain a lot of information about molecular structure, the link between an NMR spectrum and all compatible molecular structures is neither simple nor unique. It is still often mandatory to make use of a chemist's prior knowledge to reduce the number of 'admissible' molecules. With increasing sizes of the molecules of interest, w...

Though NMR spectra contain a lot of information about molecular structure, the link between an NMR spectrum and all compatible molecular structures is neither simple nor unique. It is still often mandatory to make use of a chemist's prior knowledge to reduce the number of 'admissible' molecules. With increasing sizes of the molecules of interest, w...

This Note presents an extension of the classical Lowe-Tarr λ/4 circuit which improves the protection of an NMR preamplifier against overload during RF pulses, eliminates the recovery time, and reduces the receiver dead-time. It is almost certain that this trick has been already used by others but, due to its simplicity, was not discussed in literat...

This educational Note presents a combinatorial identity and its proof, based on a practical problem encountered in the quantum mechanics of systems of n coupled spin-1/2 nuclides. Since the identity has been discussed in the sci.math group on Google, a brief comment on the discussion is also included. The author is of the opinion that even though p...

This is a clarification of the complex history of an approximate formula for ellipse perimeters that was re-discovered many times and that we propose to name simply the "particularly fruitful approximation".

We present a novel approach to the nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) analysis of complex samples with nontrivial distribution of longitudinal relaxation rate R(1). Parametrically enabled relaxation filters with double and multiple inversion (PERFIDI) aim to separate signals arising from components with different...

This Note discusses a particular set of inequalities applicable to n-tuples of positive real numbers. Despite the simplicity and potential usefulness of these inequalities, they appear to be either novel or else impossible to find in the literature (at least I did not manage). Whichever is the case, this should make the present exposition useful. H...

Biological systems often display multiexponential transverse relaxation times, diagnostic of the presence of different proton pools within the sample. A reliable representation of sample multiexponentiality can be calculated through the inversion of CPMG magnetization decays via the UPEN protocol. However, if a sample contains both liquid- and soli...

The starting point of this Note is the evaluation of the Fourier transform integrals of n-dimensional spheres with uniform density. This is closely related to the practical problem of k-space images of spheres in 1D, 2D and 3D imaging [1]. For the one-dimensional case, the answer is the well known and ubiquitous function sinc(x) = sin(x)/x which cr...

Phase correction of high-resolution NMR spectra is a routine data processing task. It used to be done by a manual adjustment of the two parameters it involves and it was left to the operator to subjectively judge whether the final result was satisfactory. More recently, due to a growing need for automatic evaluation procedures, phase-correction has...

Field Cycling NMR is a well-established and reliable method for measuring the spin-lattice relaxation time T1 and its dependence on the magnetic field strength. This is also the only NMR method capable to extend these measurements down to very low field. Unfortunately, because of instrumental limitation, it was impossible till now to apply the same...

This Note updates the following documents by the author (readers not acquainted with the topic should first consult them): (i) 2005 Review of Approximations of Ellipse Perimeters, (ii) 2006 Addendum to the Review, and (iii) 2006 Article on Algebraic Approximations of Ellipse Perimeters. ----- It also contains three apparently unrelated themes which...

Special Issue: International Symposium on Macromolecular Chemistry, Brussels-Louvain 1967 Part 2Volume 22, Issue 2, pages 1073–1084, 1969

The structure of poly(methyl methacrylate) prepared by anionic polymerization was followed. Compounds of lithium, sodium, and potassium were used as initiators. The influence of the structure of the initiator and its amount and of the monomer concentration, conversion, and temperature on the structure of the polymer in the polymerization without th...

NMR spectra of the meso and d,l isomers of methyl-2,4-pentanedicarboxylate and of isotactic, heterotactic, and syndiotactic methyl-2,4,6-heptanetricarboxylate have been measured and analyzed. From the values of vicinal coupling constants the energy differences between various conformational structures of these isomers were estimated. The limitation...

We have found a good correlation between the iodine number of maize oil and an empirical parameter-related to the proton T1 relaxation time determined by means of low resolution pulsed nuclear magnetic resonance spectroscopy. This result opens a way to fast and reliable determination of the iodine number during oil hydrogenation.

This educational article started as a response to a query in my NMR blog. The challenge was to write a brief, one-page introduction to MR for a newcomer with little or no training in physics, making its principles, main branches and technological aspects clear enough to correctly answer, for example, the question of why MR requires exceptionally st...

This Poster presents two new families of sequences developed within two different projects at the University of Bologna. The two families are related by a common feature, which is the use of a series of inversion pulses as a preamble to a classical NMR technique. The exclusive use of inversion pulses confers them a remarkable insensitivity to offse...

Quantification of proton FID signals in time-domain NMR is mostly done by fitting the experimental data to a discrete number of suitable decay functions while ignoring any frequency differences between their individual components. The latter simplification is justified when measuring systems with inherently broad lines (solids, gels, tissues, etc.)...

In NMR studies of complex systems containing many physically and chemically distinct components, one of the principal obstacles is the linearly-additive overlap of the respective signal contributions. Post-acquisition separation of the signals corresponding to the various components is possible only in some cases. The best known fortunate case is t...

An educational article that presents a slightly amusing simple instance of variational calculus. It also discusses some of the applied aspects of the presented problem which purports to regard the foundation of would-be gravity engineering, intended as a branch of space engineering. To appreciate it, you must think in terms of building large mansio...

This Note builds on the previous work by the Author as well as the recent advances made by W.Cantrell. It starts from stating the obvious, namely that any purely algebraic approximation (i.e., one which uses only the four basic operations of the real numbers algebra) to the ellipse perimeter can be written as a rational function in terms of the ell...

Using the methods developed in a previous article on field noise propagation into FID's and 1D spectra, this study investigates the effects of various types of field fluctuations on Hahn echoes and CPMG echo trains.

This educationsl Note presents an explicit formula for the exponential integral transforms of continuous functions composed of linear segments (polygonal- or P-functions). The result is then applied to the case of P-functions defined over a uniform grid of coordinates (UP-funtions) - a case in which it undergoes a considerable simplification. Apart...

This largely educational Note focuses primarily on finite sums of the power series whose terms are k^p * x^k, where p is an integer and k = 0, 1, 2, 3, ... There is nothing particularly original about this except, I hope, some aspects of the exposition. The fact is that in my work I have occasionally needed some of the identities listed in this Not...

Binary iterated powers (bips) are defined as functions of the type f(x) = β^(e_0β^(e_1β^(e_2β^(...β^(e_n x)...)))) ≡ {b_0b_1b_2...b_n|x} ≡ {b|x}, where β is a non-negative base, e_k = 2*b_k - 1, and b = {b_0, b_1, b_2, ... , b_n} is a binary sequence of 0's and 1's. This paper explores the properties of both finite and infinite binary iterated p...

NMR (Nuclear Magnetic Resonance) FID instabilities due to the most common types of magnetic field and/or RF offset fluctuations are investigated. The study analyses quantitatively some of the artifacts due to such instabilities. ----- The effect of a random, normally distributed field noise on averaged FID's is shown to be their multiplication by a...

This document updates the 2005 Review of Approximations of Ellipse Perimeters (http://doi.org/10.3247/SL1Math05.004), adding a new formula due to Cantrell. Readers who are not acquainted with the topic should consult the original review before reading this document.

This Note stresses the often overlooked fact that every finite-dimensional Hilbert space H of quantum mechanics can be cast as a spin space and that the Liouville space L of linear operators on H can be algebraically generated from just three spin-like operators (of which just two are independent). There is a considerable freedom in the choice of t...

This educational note reviews formulae suitable for the description of the effects of an RF pulse and of a free-induction delay interval under the assumption that classical, purely phenomenological description is appropriate (the same assumption which limits also the applicability of Bloch equations). This Note will be used as a reference in subseq...

The author has often encountered difficulties making students grasp some of the less evident characteristics of the Taylor power series. In particular, it is sometimes difficult to make it clear that a function f(x) may, at a given point of its domain, possess derivatives of all orders and give rise to a convergent formal Taylor series, and yet the...

Since the earliest days of Nuclear Magnetic Resonance (NMR), it has been clear that relaxation mechanisms were to play an important role in all its applications. The advent of the Fast Field Cycling (FFC) instruments has opened a number of important application areas (molecular dynamics of liquid crystals, paramagnetic contrast MRI agents, proteins...

This is a Review of various approximations to the ellipse circumference S(a,b) which, as is well known, involves the elliptic integral of the second kind E(x) and therefore can not be expressed in a simple algebraic way. Every ellipse perimeter approximation induces also an approximation for E(x) and the two approximations have the same relative er...

In the practice of electronic engineering, the noise characteristics of active devices are specified in two apparently different ways. One uses the equivalent input noise while the other, limited to devices with pre-defined input impedance, uses the concept of noise figure. There is a rather wide-spread tendency to confuse and/or misinterpret the t...

The question of NMR sensitivity is again in motion and we witness fresh attempts at improving the S/N ratios. In NMR Spectroscopy (MRS) and Relaxometry (MRR) such approaches include the use of digital receivers, dual preamplifiers, dual induction coils, cooled coils and cooled preamplifiers. ----- Some of this activity in MRS is a spin-off from NMR...

This Note reports a negative attempt to measure longitudinal relaxation 1H-NMRD profiles of solid, polycrystalline urea. Such measurements appear to be ruled out by an unusually strong piezoelectric response of urea. Moreover, the experimental results can not be explained as a simple disturbance superposed over 'normal' NMR signal since the latter...

This is the second Note in a series [1] dealing with volume and surface integrals over n-dimensional ellipsoids. Here we evaluate integrals of products of powers of the Cartesian vector components over surfaces of n-dimensional, real spheres which can be expressed by explicit formulae (this, of course, includes the surface areas themselves). -----...