Friday, July 29, 2022

Looking At Some Basic Operations & Computations In Tensor Algebra

 Basic tensor algebra is a crucial first step in getting to use tensor calculus and also a better grasp of differential geometry. (Stay tuned for Part 6 in the introductory series.) Let's begin by noting that just as in linear algebra, e.g.

 matrices factor into this.  I.e. every symmetric tensor will have the form (using dummy indices i, j):   

i j =

(a11 ……a12………a 13)

(a12….. a22……….a23)

( a13……a23………a33)

The anti-symmetric or skew symmetric tensor will have the form (note how the order of the dummy indices changes):     a j I =

(0 ……a12………-a 31)


( a31……- a23……0)

Or effectively only three distinct components.

The Kronecker delta, which we already saw in Introduction to Differential Geometry, Part 1, is:

d ik = 



(0….0… ..1)

Example 1:  Compute:  ai j  d i j   

If ai j  is a second order tensor with matrix:

Then we have:

And we let   x  i  and   y j   be first order (rank 1) tensors (i.e. vectors) respectively given by:

x i    =  (2, 1, 4)

y j    =  (3, 7, -1)

a)Find:   a ij  x i     +  a ji 

b)Find:  ( ai j    -  2/3  d  ij   )


a) From Mathcad matrix computations:


Diagonalizing tensors is analogous to obtaining the eigenvalues for a matrix in linear algebra.  Hence, we need to extract the eigenvalue equations via diagonalization and obtain the distinct eigenvalues.   Consider the object (solid tetrad) shown below for which we want to obtain  the principal axis.

For which we define a i j = 

With T’ =  A × I × At

Where t denotes the transpose.  Then we obtain, T’ =

Which is to be diagonalized.  Writing this out:

(15 - l…..0……..0)

(0…….11- l ....-3Ö2)

(0……..-3Ö2…8- l )

This leads to a cubic equation with triple roots which are:

l1 = 15,  l2 =  5, and l3 =  14

Substituting l1 in the matrix we get:

 Now find the remaining two eigenvalues, l2 and l3, to diagonaliize the matrix and obtain solutions in c x, c y and c z.

Thereby obtain the principal axes in terms of: e^’x,        e ^’y and e^’ z .

Show in particular that:

  e^’ z    = - 1/ Ö3   e1^ -  1/ Ö3   e2^ + 1/ Ö3   e3^


Using  the eigenvalue l2  = 5 we arrive at:

For which:

c y = c z / Ö 2    and  

c = 0 e1^  -  c z / Ö 2    e2^ + c e3^


(c z  2 / 2  +  c z  2)  = 1    Þ   c z   =  Ö (2/ 3)


e2^’  =

 1/Ö 2   (Ö (2/ 3)) e2^”  +  Ö (2/ 3) e3^”  = 1/ /Ö 3

e2^”  + Ö (2/ 3) e3^” 

For  the eigenvalue  l3  =  14:

We arrive at:

Yielding:   c y = -  Ö 2 c z   and   c z    =  1/ Ö 3

Þ   c   = 

Ö 2  c z   e2’’’   +  c z   e3^”’ 

=  -Ö (2/ 3) e3^”  + Ö (1/ 3) e3^”  

Then  e jå3 i= 1   a i j   e j ’ and:  e i’ =  å3 j= 1   a i j   e j


e x^”    =  - 1/ Ö 2 e 1   - 1/ Ö 2 e 2   


e x^’    =  e x^”    =  1/ Ö 2 e 1   - 1/ Ö 2 e 2    + 0


e y^”    =   1/ Ö 2 e^ 1   +   1/ Ö 2 e^ 2    + 0

And:  e z^”    =   e^ 3     Þ   e y^’ = 

1/ Ö 3 (1/ Ö 2) e^ 1   +  1/ Ö 3 (1/ Ö 2) e^ +   Ö (2/ 3) e3

=   e^ 1   / Ö 6   +   e^ 2   / Ö 6   +   2e^ 3  / Ö 6  

And:  e z^’    =  

 - 1/ Ö 3  (e^ 1 ) - 1/ Ö 3  (e^ 2)   + 1/ Ö 3  (e^3)  

(The Principal axes are: e x^’ ,    e y^’   and  e z^’ )

Suggested Problem:

In a certain rectangular coordinate system, the directions of whose axes are given by the unit vectors i, j and k, the inertia tensor of an object is given by:

I = K x= 



(0….1… ..1)

a) What are the principal moments of inertia of the object (the moments of inertia along the principal axis) relative to the origin of the above coordinate system?

b) What is the direction of the principal axis corresponding to the principal moment of inertia and equal to K?

c)If the origin of the above rectangular coordinate system is at the center of mass of the object and the total mass of the object is M, what is the change in the inertial tensor of the object if the rectangular coordinate system is displaced parallel to itself a distance ro in the direction


 (1/ Ö2)j +  (1/Ö2)k?

PSMA Scan Confirms Cancer Is Metastatic (As Well As Localized) - And Urgent Action Is Needed


Previous axumin scan taken just over a year ago - showing main nodes.

PSMA scan showing same region taken on June 8th

The recent death of actor Tony Dow was again a reminder that once we reach the 70s (Dow was 77) the medical problems grow and may become catastrophic.  In a NY Times piece yesterday (Big brother on “Leave It to Beaver” dies) by Anita Gates we learned:

Dow had been in hospice care and announced in May that he had been diagnosed with prostate and gall bladder cancer.

 The case of metastatic cancer of any form is always bad news, as my youngest brother Mike learned when diagnosed in April, 2018 with metastatic liver cancer e.g.

Such outcomes are seldom good, and Mike lasted barely 2 months after his cancer was missed at a VA center. (Later diagnosed by a Medicare -registered physician.) He declined the use of radiation and chemotherapy to extend his life "another few months", opting to get it over with asap.  In Mike's case, his choice was in line with author Barbara Ehrenreich's arguments (in her book:  'Natural Causes: An Epidemic Of Wellness, The Certainty Of Dying, And Killing Ourselves to Live Longer') . I.e.  that after a point it makes no sense to fight on, even for a few months, e.g.

This problem of choice - when to let go - is obviously going to depend on the person's cancer - or other medical condition-  their personality, as well as belief system, if any.  Atheists - like me and Barbara Ehrenreich - are more likely to take the philosophical view below, expressed in her book:

"You can think of death bitterly or with resignation, as a tragic interruption of your life, and take every possible means to postpone it.  Or, more realistically, you can think of life as an interruption of  an eternity of personal nonexistence, and seize it as a brief opportunity to observe and interact with the living, ever surprising world around us."

For me, after examining  the PSMA scan done back on June 8th, e.g.

Better Than The Axumin Scan Last Year- The PSMA

 (which for some reason was delayed 3 weeks in getting to me), it may be true that the cancer is ready to wreak more havoc in the prostate itself as well as beyond. Before revealing the specific findings let's bear in mind the PSMA uses a more sensitive radionuclide than the axumin agent given, in this case something called  F-18 Pilarify which:

 "Binds to prostate cancer cells to help localize prostate cancer cells."

It is therefore the property of the agent's binding to specific prostate cancer cells that validates and empowers its diagnostic use in scans.  The cancer cells then will experience measurable uptake of the agent which helps to identify them.  That uptake is referenced in the accompanying scan report, i.e.:

 "There is increased uptake in the mid to right posterior aspect of the prostate gland extending into the anterior aspect of the prostate gland in the midline and left of midline with a peak SUV of 8.7 consistent with prostate cancer."

The report adds:  "There is increased uptake in bilateral posterior external iliac lymph nodes.  The most intense uptake is seen in a left 9  x 20mm lymph node at the level  of the roof of the left acetabulum with a peak SUV of 5.1 (image 51)  Increased uptake is also seen in bilateral small distal external iliac lymph nodes.

Noting also:  "This is consistent with metastatic prostate cancer."

So we know two things: a)  Localized 'intense' nodes appear in the gland itself and b) the cancer is already in the lymph nodes and hence, metastatic. This means to me that a metastatic solution - treatment is needed, not more radiation. Which is a terrible idea anyway after a patient has already had high dose radiation as a primary treatment. 

 Let me also say a few words about SUV.  This means: 'Standardized Uptake Value, and in the radiologic literature is defined thus:

 As the name suggests it is a mathematically derived ratio of tissue radioactivity concentration at a point in time C (T) and the injected dose of radioactivity per kilogram of the patient's body weight: SUV = C (T)/ [injection dose (MBq)/patient's weight (kg)]

SUV = C(T)/[injection dose (MBq)/patient's weight (kg)] 

It is used to measure response of cancers to treatment and is considered a semi-quantitative value as it is vulnerable to other sources of variabilities. The most reliable method of measuring activity levels would be to use fractional uptake rate (FUR) which is measured from blood samples. As the FUR and SUV are proportional, related to plasma clearance rate and a dimensionless initial distribution volume, careful usage of SUV is widely used over blood sampling.

SUV may be influenced by image noise, low image resolution and variable user-biased region of interest (ROI) selection. The cut off between benign and malignant lesion/nodule is in the SUV range of 2.0-2.5. PET sensitivity and specificity decreases with lesions smaller than 7 mm. Thus, continued follow-up is recommended. It is important to note that many infectious and inflammatory processes will also have high SUV.

If any good news appeared in the report it was that no bone mets were found.  Nevertheless, the spread to the lymph nodes means some essential action needs to be taken now and that is likely to be with the newest ADT (anti-androgen)  drug (Orgovyx). 

Recall the job of all the ADT regimens (whether tri-monthly Lupron shots or daily Orgovyx pills)  is to radically slow the production of  ' T ' or testosterone. This is given it is the T which feeds the cancer cells and enables them to metastasize, wherever they are.  This 'medical castration' approach  was first developed in the 1940s, but remains the last and literally only hope for men with advanced prostate cancer. (The other is surgical castration or orchiectomy which we are told is vastly cheaper than the drug route - but fewer get it for obvious reasons.)

Anyway, a visit with the urologist next week (and likely 3 weeks later than it should have been) will enable me to learn how soon I can get the necessary prescription. Why get it if I am philosophical re: end of life issues?  Well, because as Janice has noted (as a radiation treatment therapist she had to treat hundreds of PCa patients in Barbados)  there are dozens of horrific things that can occur well before the end., like the cancer breaking through the bowel requiring an ostomy bag.   You definitely want to delay those as long as you can and ADT - especially with Orgovyx - offers a way to do that.

Also, it would be terrific if a Senate bill to lower the costs of prescription drugs offered via Medicare can help me to save a bundle.    (We have to hope Kyrsten Sinema doesn't scuttle it at the last minute.)  We will see!  

See Also:

PSMA PET Scan for Prostate Cancer | UCSF Radiology


Wednesday, July 27, 2022

Lab Experiments (Especially Using Plasma) Needed To Assess The Tetrahertz Gap In Astro Chemistry

 It is a fair conclusion that no one who attended the astrochemistry lecture by Paolo Caselli at the American  Astronomical Society 236th meeting in 2020, e.g.

Ever heard of the "tetrahertz gap" in the radio portion of the electromagnetic spectrum.   Which suggests either that this gap is of no compelling importance, except to the few astrochemists -astrophysicists who work in this area, or that the academic discipline itself lacks sufficient resources and empirical  'objects of inquiry' to pursue at a very rigorous level.  Never mind.  We have Prof. Sussana Widicus Weaver, based at the Univ. of Wisconsin =Madison, to thank for setting the record straight. (At least from her frame of reference!)  

This was in her recent Physics Today essay,  Astrophysics in the Tetrahertz Gap. 

In terms of relevance and importance, Prof. Weaver reminds us that "studies of the atomic and molecular universes rely heavily on various spectra recorded in the long-wavelength range of the electromagnetic spectrum. At those wavelengths, astronomers identify the fingerprints of organic molecules, determine the conditions inside stellar nurseries, and detect redshifted transitions of atoms in distant galaxies." The subjects include complexes of molecular clouds and star-forming regions in the Milky Way.  See e.g. some introductory material here:

Much of the previous work, Weaver notes,  "has been conducted in the microwave regime, which covers 300 MHz to 300 GHz in frequency, or 1 m to 1 mm in wavelength. But recent insights into the molecular universe have come from far-IR (FIR) observations in the range of 300 GHz to 20 THz in frequency, or 1 mm to 15 μm in wavelength."
 The era of far infrared astronomy was brought about by the Herschel Space Observatory, in tandem with the Atacama Large Millimeter/Submillimeter Array (ALMA), and the Stratospheric Observatory for Infrared Astronomy (SOFIA). Together we're told these have  "led to a recent heyday of molecular astronomy."
But still, there's this gap.  Weaver elucidates (ibid.):
"In the past, advancements in this field have been held back by the “gap” in the terahertz regime arising from the relative lack of molecular spectroscopic information in this range as compared with other regions of the electromagnetic spectrum. The gap, illustrated in figure 1, had arisen because of historical limitations in the technology available for laboratory and observational studies.
                                             Fig. 1. Tetrahertz gap in radio spectrum

 Because of advances in telecommunications, security, and astronomical instrumentation design in the past 20 years, however, astronomers are now making rapid improvements. New tunable terahertz light sources, higher-power terahertz amplifiers, more sensitive detectors, and rapid and broadband data-acquisition capabilities have revolutionized the field. Researchers are starting to fill the terahertz gap."
This brings up the properties of molecular spectral lines. But what is a basic spectral line? Atoms as well as molecules absorb only at certain wavelengths that correspond to specific energy transitions.  For example I show some specific transitions of spectra in the diagram below:

As shown in the diagram,  emission occurs when an electron in the atom, say hydrogen, makes a transition from a higher to a lower energy level. This is always  accompanied by the emission of a photon with a defined energy E = hf = h (c/ l).  For both emission and absorption multiple aspects may need to be considered, including transition probabilities. See e.g. my earlier post:

 For molecules, as Prof. Weaver observes, the line production can also arise from energy changes associated with rotation, vibration, or electronic energy. Molecular transitions are quantized, which simply means that the transitions occur only at particular amounts, or quanta, of energy. The transitions are sharp, meaning that they happen over a narrow range of frequencies.
What I hadn't been aware of previously is that a number of important features have been identified already within this "FIR" regime including  the electron spin flip of the hydrogen atom at a wavelength of 21 cm; the ammonia structural inversion transitions at a frequency of 23 GHz; the pure rotational lines of carbon monoxide, which are the signposts of telescope receiver bands; and the rotational and rovibrational lines of organic molecules that have been seen in comets and the interstellar medium and may be the precursors to life throughout the universe.
That's a lot to pack in as well as appreciate.
 Molecular spectral transitions appear as narrow spikes (hence the use of the word “line” to describe them). Each line in a spectrum corresponds to a specific transition of the molecule. Figure 2 below shows an example molecular spectrum as shown in Weaver's PT article:

 The specific energies of rotational transitions are related to a molecule’s structure. As such, each rotational spectrum is molecule specific. If several lines for a given molecule are observed, the temperature and density in a sample of gas can be quantified by comparing the lines’ relative intensities As Weaver goes on to note:
"Molecules, therefore, are routinely used as probes of the physical properties of objects in space. For astrochemists, the molecular information gained through such measurements improves their understanding of how chemistry evolves as stars and planets form.
To study a particular molecule in space, one must first obtain a laboratory spectrum and assign the lines therein. Spectral lines are measured across a given frequency range and matched to a spectral prediction based on a quantum mechanical model of the molecule’s energy levels. Usually not all of a molecule’s spectral lines are measured."

This is in line with what I already discussed in the previous blog post link. Further we know transition probabilities, Frequency coverage and sample conditions all influence what lines can be observed in the lab and in space. But even with only partial spectral information from the laboratory, one can assign the spectrum and predict the rest of the lines.  Also important to note:
"Simple molecules require only a few parameters for a full spectral assignment. Complex molecules, however, demand dozens of parameters to achieve a level of assignment that leads to reliable predictions. If researchers collect sufficient information in the lab, they can determine the parameters with a high degree of precision. They can then use the information to extrapolate the spectrum to other frequency ranges at any temperature.
The spectrum that is collected during telescope observations contains all the lines from every molecule in the source. When a molecule is identified, its spectral features are matched to those predicted from the laboratory spectral assignment. Matching requires knowledge of not only the spectrum for the molecule of interest but also the spectra for all the molecules in the source. 
Astrochemists can then connect each line to a molecule and sort out any ambiguities by recognizing the patterns from known laboratory measurements. If enough spectral lines are observed that can be uniquely assigned to a given molecule and if their relative intensities match the expected physical parameters of the source, the molecule is said to be detected in space."
But again, the key question is the extent to which we can have confidence that such a process yields laboratory spectra that can be matched to actual astrophysics observations. And if not, do we really need laboratory spectra? Unfortunately, laboratory studies have been limited because of the challenges of filling the terahertz gap. As Weaver points out, historically, "researchers have lacked stable, high-powered, tunable light sources in the far infrared regime and also lacked sufficiently sensitive detectors to cover that part of the electromagnetic spectrum". 
Infrared (IR) spectroscopy itself, covering 20–430 THz in frequency, or 15 μm to 700 nm in wavelength extent, is well established.  After all, commercial spectrometers are a standard tool in nearly every chemistry laboratory. Microwave spectroscopy, covering 300 MHz to 300 GHz in frequency, or 1 m to 1 mm in wavelength, is not as widely used as an analytical tool. But it is just as powerful as IR spectroscopy and is a well-established field of research dating back to the development of radar during World War II.
Weaver's contention is that far infrared spectroscopy has not been widely pursued. Of all the high-resolution spectroscopy research laboratories in the world, only around 10 have spectral access from 300 GHz to 1 THz. The number of labs with high-resolution spectral access above 1 THz can be counted on one hand. But with the development of new FIR observatories has come new technological capabilities, and many of the historical limitations in the FIR range have been overcome. A deluge of astronomical spectra is now arriving from FIR telescopes. In Weaver's words:
 "My research group and the handful of others who work at the millimeter-to-micrometer wavelengths are striving to develop laboratory techniques to keep up with the quantity of data."
But then there are many other complexities at hand, most of which arise from having to use 'proxy' spectral bands. Weaver admits, for example, the FIR methods have to draw from both the infrared and microwave regimes. Indeed, in the former realm, we are informed "light sources that include lasers and optics based on ground glass with mirrored coatings are used" while in the latter, "radiation is generated by crystal oscillators, like those used in a watch and car radio".  For this reason we are told "the FIR is often called the quasi-optical regime because it draws methods and equipment from both approaches. Combining the two types of experiments into one system that works for all wavelengths in the FIR is complicated and technologically challenging."
Other complexities and challenges cited by Weaver (ibid.);
- Once a system is established for generating light and directing it into the sample, a detector sensitive enough to measure the signals is required. But anything above roughly 300 GHz requires a custom-built detector cooled with liquid helium. (Unless kept at extremely low temperatures, the detector elements that have the best response at FIR wavelengths are flooded with thermal background noise.)
- Helium boils at a temperature of 4.2 K, and keeping a detector at that temperature requires routine cryogen fills, which greatly complicates the logistics of experiments. 
- The natural supply of helium on Earth is rapidly dwindling, and using helium-dependent devices is increasingly expensive (see Physics TodayApril 2019, page 26, and “Helium shortage has ended, at least for now,” Physics Today online, 5 June 2020). (Detector manufacturers are now implementing closed-cycle cooling systems that use helium recirculation to better conserve the supply. But those detectors are not readily available to every spectroscopy laboratory.)
- Once a source and detector are set up, the next step is to deliver the molecular sample into the system....For molecules that are highly reactive or unstable, however, they need to devise ways to produce the molecule and keep it sufficiently isolated to avoid its reaction or decomposition while recording spectra. Most options require sources that continually flow or pulse the gas mixture through the system.
- Those sources introduce complications for gas handling because they require large pumps and vacuum fittings, windows, and other hardware to couple the spectrometer to the gas cell.
-Another challenge has to do with the nature of the molecules. In the low density of space, molecules react slowly because they take tens of thousands of years to collide. 
Weaver acknowledges that using ions - say from lab plasmas - can speed up the process experimental considerably. Trouble is,  ions are difficult to produce in the lab at sufficient quantities to study their spectra. While plasmas can be used to make the ions (and then the gas sample is expanded into a vacuum) there isn't a sufficient density to ensure success. Even with the most efficient ion sources, only 1 in every 10 000 molecules gets ionized. Once expanded into the vacuum, the gas sample is even further diluted.
It turns out that detecting some ions in the lab is more difficult than doing it in space which makes an outside observer wonder why bother?  But Prof Weaver makes a decent case that - at least for the time being- and for the benefit of future astrochemistry in the tetrahertz range, we need to use proxy methods as an accompaniment to spectrometric space observations.