aBa's Webpage
Office: Rm. 526 in Hibbard Hall
Email: mbirika@uwec.edu
- Address:
- aBa (אבא) Mbirika
- University of Wisconsin-Eau Claire
- Department of Mathematics - HHH508
105 Garfield Avenue
Eau Claire, WI 54701
USA
Current Position
(2022 to Present) -- Professor at University of Wisconsin-Eau Claire
(2017 to 2022) -- Associate Professor at University of Wisconsin-Eau
Claire
(2013 to 2017) -- Assistant Professor at University of Wisconsin-Eau
Claire
Previous Positions
(2010 to 2013) -- Postdoctoral Fellow and Instructor at Bowdoin College (in
Brunswick, Maine)
(2004 to 2010) -- Graduate Teaching Assistant at University of Iowa (in
Iowa City, Iowa)
Education
Ph.D. in Mathematics (2010) -- University of Iowa
M.S. in Mathematics (2006) -- University of Iowa
B.A. (2004) -- Sonoma State University
Professional Documents
CLICK
HERE to see links to my Associate Professor Promotion and Tenure
Binders, CV, Research and Teaching Statements, Class Evaluations, etc
Do you want to ask me to write a Letter of Recommendation for you?
Then
first CLICK HERE and
read this advice in its entirety before requesting.
Teaching (at University of Wisconsin-Eau Claire)
Fall 2023 - Math 104 (Section 001 and 002) - Finite Mathematics - syllabus and constantly updated Notes Available Here
Fall 2023 - Math 425 - Abstract Algebra I - syllabus and constantly updated Notes Available Here
Fall 2023 - Research on Narayana and various other sequences with Morgan
Fiebig and Dr. Jürgen Spilker (emeritus faculty of University of Freiburg,
Germany)
CLICK
HERE to see an exhaustive list of all the classes I taught from 2013
to the present at UWEC. This may be useful
to prospective students as these contain links to former course
syllabi.
Previous Teaching 2010 to 2013 (at Bowdoin College)
CLICK HERE to see an
exhaustive list of all the classes I taught from 2010 to 2013 before
teaching at UWEC.
Teaching Awards at UWEC
2022 - UWEC Certificate of Recognition for Contributions to Students (32
total awarded campus-wide) - click here to see award
2016 - UWEC Certificate of Recognition for Contributions to Students (39
total awarded campus-wide) - click here to see award
2014 - UWEC Certificate of Recognition for Contributions to Students (22
total awarded campus-wide) - click here to see award
Research Publications
- Pell and associated Pell braid sequences as GCDs of sums of k
consecutive Pell and balancing-related numbers. (Joint work with
Janeč Schrader and Jürgen Spilker) -- Published by Journal of Integer
Sequences, 26 No.6, Article 23.6.4 (2023), 24pp.
(Available by clicking here).
- GCD of sums of k
consecutive squares of generalized Fibonacci numbers. (Joint work
with Jurgen Spilker) -- Published by Fibonacci Quarterly, 60
No.5 (2022), 255--269. (Available by clicking here).
- GCD of sums of k
consecutive Fibonacci, Lucas, and generalized Fibonacci numbers.
(Joint work with Dan Guyer) -- Published by Journal of Integer
Sequences, 24 No.9, Article 21.9.8 (2021), 25pp.
(Available by clicking here). (This arXiv version is easier to navigate with better
numbering system.)
- New methods to find patches of
invisible integer lattice points. (Joint work with Austin
Goodrich and Jasmine Nielsen) -- Published by Involve: A Journal of
Mathematics, 14 No.2 (2021), 283--310. (Available by
clicking here).
- Representation stability of Springer
varieties and some combinatorial consequences. (Joint work with
Julianna Tymoczko) -- Published by Journal of Algebraic Combinatorics, 53
No.3 (2021), 897--920. (Available by clicking here).
- An Euler phi function for the
Eisenstein integers and some applications. (Joint work with
Emily Gullerud) -- Published by Integers: Electronic Journal of
Combinatorial Number Theory, 20,
Paper No. A20 (2020), 28pp. (Available by clicking here)
- Lattice point visibility on
generalized lines of sights. (Joint work with Edray Goins,
Pamela Harris, and Bethany Kubik) -- Published by the American
Mathematical Monthly, 125 No.7
(2018), 593--601. (Available by clicking here).
- Coprime and prime labelings of
graphs. (Joint work with Adam Berliner, Nate Dean, Jonelle
Hook, Alison Marr, and Cayla McBee) -- Published by Journal of Integer
Sequences, 19 No.5, Article
16.5.8 (2016), 14 pp. (Available by clicking here).
- On the sign representations for the
complex reflection groups G(r,p,n). (Joint work with Thom
Pietraho and Bill Silver) -- Published by Beitrage zur Algebra und
Geometrie, 57, Issue 4 (2016),
851--858, DOI 10.1007/s13366-016-0289-3, (Online published version
available by clicking here).
- Generalizing Tanisaki's ideal via
ideals of truncated symmetric functions. (Joint work with
Julianna Tymoczko) -- Published by Journal of Algebraic Combinatorics, 37 (2012), 167--199.
(Available by clicking here).
- A Hessenberg generalization of the
Garsia-Procesi basis for the cohomology ring of Springer varieties.
Published by Electronic Journal of Combinatorics, 17
No.1, Research Paper 153 (2010). (Available by clicking here).
Accepted (and awaiting
publication details)
- Tantalizing properties of subsequences of
the Fibonacci sequence modulo 10. To appear in Rocky Mountain
Journal of Mathematics. (Joint work with Dan Guyer and Miko Scott)
(arXiv version available by clicking here)
Submitted
Preprints (Ready to submit):
- Tridiagonal real symmetric matrices with a connection to Pascal’s
triangle and the Fibonacci sequence. (Joint work with Emily
Gullerud and Rita Post) (arXiv version available by clicking here)
In Preparation:
- Towards a Schur-Weyl duality for the
alternating group. (In progress) (Available by clicking here).
- Involutions in the hyperoctahedral
group. (Joint work with Rita Post and Ryan Stodola)
Contributions to the On-Line
Encyclopedia of Integer Sequences (OEIS):
- A213273 (sequence): The smallest m
such that the complete bipartite graph K_{n,n} has a coprime labeling
using labels from {1,...,m}. Available by clicking here.
- A210219 (paper and formula links): GCD of all sums of n
consecutive Fibonacci numbers. Available by clicking here.
- A229339 (paper and formula links): GCD of all sums of n
consecutive Lucas numbers. Available by clicking here.
Interdisciplinary Research Publications
- The Three Giri of Paradiso
XXXIII. (Joint work with
Arielle
Saiber). Published by the journal, Dante Studies, 131
(2013): 237--272. (Available by clicking here).
Expository Math Writings
- Hidden
trees in the forest: On lattice points and prime labelings of graphs.
This paper was inspired by my one-week residency at an AIM-NSF
workshop called "Research Experiences for Undergraduate Faculty" at
ICERM in Providence, RI, in June 2012. It contains some
of our unpublished results, and also includes exercises and project
ideas that faculty can use to involve undergraduates in research in the
areas of Combinatorics, Graph Theory, and Number Theory (2012).
(Available by clicking here).
- On a coin-flip problem and its
connection to pi. I wrote this with the intention of
using it as a teaching (learning) tool for other faculty (students).
Hence many exercises are included (2011). (Available by
clicking here).
Textbooks and Course Packets (self-published and public domain)
This textbook was written by myself and co-author Rita Post over the Summer
2018 and Fall 2019 semester. It was first used in my Spring 2019 number
theory course Math 341 at UWEC. The book has two forms as follows:
- The Power of Proofs [AND]
Number Theory. (Joint work with Rita Post).
Self-published with Creative Commons License (CC BY-NC-SA), 2019.
The following course packet was written by me for my students in Math 104.
The first version was used in my Fall 2020 class.
- Finite Mathematics - The
Power of [Math 104]. Self-published with Creative
Commons License (CC BY-NC-SA), 2022.
The following course packet was written by me for my students in Math
314. The first version was used in my Fall 2022 class.
- Discrete Mathematics - The
Power of [Math 314]. Self-published with Creative
Commons License (CC BY-NC-SA), 2022.
The following course packet was written by me for my students in Math
425. The first version was used in my Fall 2021 class.
- Abstract Algebra I - The
Power of [Math 425]. Self-published with Creative
Commons License (CC BY-NC-SA), 2021.
Variety of Useful Resources written by my students
NOTE/WARNING: Some of the documents below are solutions
to exercises in some textbooks. These may not be error-free, so if you are
a STUDENT "borrowing" solutions from these documents, you
should NOT be doing that. You should work on your own solutions
first and then maybe glance at what you find here.
- A Handy Guide to the Basics
of LaTeX. Author: Dan Guyer. Self-published with
Creative Commons License (CC BY), 2021.
- The following was written by students and myself in my Spring 2020
course on primitive roots and quadratic reciprocity. Authors: Jonah
Amundsen, Dan Guyer, Bob Langer, aBa Mbirika, and Maggie Reardon (Spring
2020)
- The following is preliminary results on the Fibonacci and Lucas
sequences identities. Author: Janeč Schrader (Summer 2022)
- The following is preliminary results in elementary number theory
identities, and Pell, Pell-Lucas, balancing, cobalancing,
Lucas-balancing, and Lucas-cobalancing identites. Author: Janeč Schrader
(Summer 2022). WARNING: Some exercises are not
complete due to Janeč and I conducting research in Fall 2022.
Current Research (this section needs updating)
Broadly speaking, I am interested in combinatorics as related to areas such
as combinatorial representation theory and algebraic geometry/topology.
My collaborator Julianna
Tymoczko (recently from University of Iowa and now at Smith College)
and I worked on a joint project on truncated
symmetric functions towards a combinatorial description of the
cohomology ring of Hessenberg varieties, which are a generalization of
Springer varieties. We recently
completed work on a project on co-FI-modules and representation stability of
the cohomology of Springer varieties. My collaborators Thomas Pietraho, Bill Silver, and I worked on a joint
paper on the sign representations of imprimitive complex reflection groups,
G(r,p,n). During my Spring
'08 semester residence at MSRI, I begun a project (currently
on hold - but would love to start again perhaps with a
research student here at UWEC) on the Schur-Weyl duality of the alternating
group begun with Tom
Halverson of Macalester College and continued with co-advisor Fred Goodman of University of Iowa.
Here are brief introductions to
some of the objects or ideas listed above that I think about:
- On my
Schur-Weyl duality project. This 2 page write-up describes
the problem that I am working on. It is also a good introduction
for an undergraduate interested in doing research with me on this
project.
- Complex
reflection groups. This 5 page write-up briefly describes
the family of imprimitive complex reflection groups (giving various
combinatorial ways of viewing their elements), irreducible
representations of these groups, a description (and an example) of
a generalized Robinson-Schensted correspondence between these groups and
pairs of same-shape multitableaux, and some combinatorial implications
of the latter map.
Research
Projects with Undergraduates at UWEC:
- Lattice point visibility:
In 2013-2014, my two students Jasmine Nielsen and Austin Goodrich
research n-by-n
square patches of invisible lattice points in the integer lattice - see
our ORSP Grant
Proposal for more info. This project was continued in an SREU
project - see our ORSP
Grant Proposal for more info.
- Generalized lattice point visibility:
In
2015-2016, my two students Sara DeBrabander and Michele Gebert research
lattice point visibility along generalized lines of sights (e.g.,
parabolic lines of sights) - see our ORSP
Grant Proposal for more info.
- Longest increasing subsequences in
G(r,p,n): In Summer 2016, my three students Rita Post,
Ryan Stodola, and Stan/Jingtai Liu research various combinatorial
statistics on elements in the complex reflection groups G(r,p,n) - see
our ORSP
Grant Proposal for more info.
- Permutation statistics on complex
reflection groups: In 2016-2017, my three students from Summer
2016 continue their work. This time we focus on involutory conjugacy
classes in the hyperoctahedral group G(2,1,n). - see our ORSP Grant
Proposal for more info.
- Characteristic polynomials and
eigenvalues for a family of tridiagonal real symmetric matrices:
From Summer 2017-Spring 2018, my students Emily Gullerud and Rita Post
investigated the spectrum of a certain class of tridiagonal matrices. We
found a tantalizing connection between this spectrum and certain
diagonals in Pascal's triangle. There is no ORSP grant proposal for this
project as we did this purely for research joy!
- Combinatorial applications of
Gaussian and Eisenstein integers: In Summer 2017, my student
Emily Gullerud and I investigate combinatorial applications of Gaussian
and Eisenstein integers - see our ORSP
Grant Proposal for more info.
- Involutory and orientation-preserving
symmetries in the hyperoctahedral group G(2,1,n): In 2017-2018,
my three students Emily Gullerud, Rita Post, and McKenzie Scanlan are
continuing work on the involutory conjugacy classes started in the
2016-2017 year with Liu-Post-Stodola. Furthermore, we investigate the
splitting criterion for the alternating group A(2,1,n) conjugacy
classes. - see our ORSP
Grant Proposal for more info.
- GCD of sums of k
consecutive Fibonacci, Lucas, and generalized Fibonacci numbers:
In Spring 2019, my student Dan Guyer and I began investigating the GCD
of sums of k consecutive
Gibonacci numbers. This unfunded research project led to a preprint that
was finalized by January 2021 and ready for submission to a journal.
There is no ORSP grant proposal for this project as we did this purely
for research joy!
- Properties of the Fibonacci sequence
modulo 10: In Summer 2019, my two students Dan Guyer and Miko
Scott investigate patterns in the subsequences of the Fibonacci sequence
modulo 10. - see our ORSP Grant Proposal for more info.
- (1) Properties of the Fibonacci
sequence modulo 10 (continued) and (2) The Catalan sequence
application to Pascal's triangle: In 2019-2020, there were two
distinct projects. The first was a continuation of the Summer 2019
project with Dan Guyer and Miko Scott. The second is a project on a
Catalan-weighted binomial sum formula that is conjectured by Dr. aBa to
give the Betti numbers of the cohomology ring of the Springer variety.
This research is conducted with students Dan Guyer, Miko Scott, and
Maggie Reardon. - see our ORSP Grant Proposal for more info.
- An analysis of primitive roots and
Menon’s Identity in the Eisenstein integer ring: In
2020-2021, my students Dan Guyer and Lily began investigating the
primitive roots in the Eisenstein integer ring by examining the units
groups of certain quotients modulo ideals generated by powers of an
Eisenstein prime. - see our ORSP Grant Proposal for more info.
- Generalizing the Fibonacci sequence modulo 10 research project
to arbitrary moduli: In 2021-2022, my students Alyssa Franks
and Javier Sanchez generalized the modulo 10 Fibonacci research project
(from Summer and Fall 2019) to arbitrary moduli settings. - see our ORSP
Grant Proposal for more info (NOTE: This
proposal was for a different project, that was completed by Dr. aBa and
Dr. Jurgen Spilker of Freiburg University in Germany before the research
with Franks and Sanchez would begin. Hence, Franks, Sanchez, and Dr. aBa
deviated from this proposal and attempted a different research
question).
Research Workshops Attended
Recent/Future Invited Talks/Conferences/Colloquiums
- ORSP Faculty/Academic Staff Faculty Forum - Title: Tantalizing Tale of
Numerical Sequences - Nov 1, 2023 - (University of Wisconsin-Eau Claire)
- Abstract
- AWM Research Symposium - Title: Tantalizing Adventures with my
Undergraduate Research Students on Numerous Numerical Sequences - Sept
30, 2023 to Oct 2, 2023 - (Clark Atlanta University in Atlanta, Georgia)
- Abstract
- UWEC Math Colloquium - Title: From Fibonacci's rabbits to Narayana's
cows: A tale of two sequences (including VERY brand new results)
- May 11, 2023 - (University of Wisconsin-Eau Claire) - Abstract
- Macalester College MSCS Senior Banquet - Title: From Fibonacci's
rabbits to Narayana's cows: A tale of two sequences - April 20, 2023 -
(Macalester College in Saint Paul, MN) - Abstract
- The Twentieth International Conference on Fibonacci Numbers and Their
Applications - Title: GCD of sums of $k$ consecutive Fibonacci, Lucas,
and generalized Fibonacci numbers - July 25-29, 2022 - (University of
Sarajevo in Sarajevo, Bosnia and Herzegovina) - Abstract
- High Point University Math Colloquium - Title: Mind-blowingly
awesome subsequences in the Fibonacci sequence modulo 10 - Sept
17, 2021 - (High Point, NC) - Abstract
- Providence College Math Colloquium - Title: Tantalizing
explorations in the Fibonacci and Gibonacci Sequences - Apr 29,
2021 - (Providence, RI) - Abstract
- STEM Summer Bridge Math Presentation - Title: My
Path to Becoming a Math Major and The Beauty of Math - July 20,
2020 - (Sam Houston State University in Huntsville, TX)
- Sonoma State University Math Festival (Main Invited Speaker) - Title:
Mathematics Research Inspired by
Astrology: The Fibonacci Sequence Modulo 10 - Apr. 22, 2020 -
(Rohnert Park, CA) - Abstract
- Winona State Math Colloquium - Title: Two
Research Projects Birthed from Curiosity, Recreation, and Joy -
Feb 28, 2020 - (Winona, MN) - Abstract
- AMS-MAA Joint Mathematics Meeting [MAA Invited Address] - Title: Two Research Projects Birthed from
Curiosity, Recreation, and Joy - January 14-18, 2020 - (Denver,
CO) - Video
of Talk
- aBa, update this and add your talks between
Jan 2018 and Jan 2020 !!!!!
- AMS-MAA Joint Mathematics Meeting [AMS Contributed Paper Session on
Matrices and Matroids] - Title: Characteristic
polynomials
and eigenvalues for a family of tridiagonal real symmetric matrices
and a tantalizing connection to Pascal's triangle - January
10-13, 2018 - (San Diego, CA) - Abstract
- Mathematics Seminar - Title: An
Euler phi function for the Eisenstein integers and some tantalizing
applications - Nov. 17, 2017 (Florida Gulf Coast University at
Fort Myers, FL) - Abstract
- AMS Central Sectional Meeting [Special Session on Algebraic
Combinatorics of Flag Varieties] - Title: Combinatorial consequences of the
representation stability of the cohomology of Springer varieties
- Sept 9-10, 2017 (University of North Texas at Denton, TX) - Abstract
- AMS Central Sectional Meeting [Special Session on Combinatorics
and Representation Theory of Reflection Groups: Real and Complex] -
Title: Involutory and
orientation-preserving symmetries in the hyperoctahedral group
- Sept 9-10, 2017 (University of North Texas at Denton, TX) - Abstract
Previous Invited Conference/Colloquium/Seminar Talks Given (from
2006--Present)
CLICK HERE to see an exhaustive
list of my talks and presentations given since 2006.
What is "Math in the Woods"?
Math in the Woods is an event that is hosted by the UWEC Math
Dept at the beginning of each Fall semester. A main goal of the event is
to have fun and build community. We host the event here in a park in Eau
Claire. All are welcome: students and their families, faculty/staff and
their families, and even our dogs too! At Math in the Woods,
you'll find a community of new friends and great company.
Here is a link to a photo journal of our most recent event: Math_in_the_Woods_Fall_2021_photo_journal.
Service (at University of Wisconsin-Eau Claire)
CLICK HERE to see an exhaustive list of
my previous service from 2013--Present.
Previous Service (at Bowdoin College and University of Iowa)
CLICK HERE to see an
exhaustive list of my previous service from 2004--2013.
Professional Affiliations
Member, American Mathematical Society -- AMS
Member, Mathematical Association of America -- MAA
Member, Project NExT
Fellow (Blue 10 Dot)
Personal Stuff
Interview
of aBa by Megan Monday of Love Wisconsin
Fun Stuff
Video Interviews of aBa (filmed at UWEC)
My Power-of-AND
video (filmed by Glen
Mabie in UWEC News Bureau Dept.)
UWEC
TV-10 Video in Fa'14 (filmed
by UWEC student Abby Reimer for BNN Blugold News Now Behind
the Desk Series)
Interview by Paige
Wilde in Sp'17 (filmed
by UWEC student Paige Wilde for a Public Relations course)
Videos of Certain Research Talks
Videos of me explaining mathematical concepts to my students via
song/rap
I wrote this for my Spring 2012 Bowdoin students in Calculus II.
Lyrics sheet available here.
This is my first ever combinatorial "poem". I wrote it for my Bowdoin
students in Spring 2011 to give them a wee break and a fun overview of the
two major topics covered in our Math 258 course. Lyrics sheets with
blackboard shots available by clicking here.
When I found out my office mate Jonas Meyer was holding analysis class on a
school holiday, I felt bummed out for the students. So I decided to
write them a math treat. The holiday was July 4th (the next day), so
there's a mix of patriotism in this poem about analysis. Lyric sheet
available here.
This poem was delivered to my Abstract Algebra class at the Univ of Iowa in
Spring 2009. I actually wrote it while I was an undergrad at Sonoma
State University in Northern California. Lyric sheet available here.
The above is a poem I wrote for the class I taught in Fall 2008 at the Univ
of Iowa. My friend Max filmed me in a Chinese restaurant in NYC with
some sorta Blackberry gizmo or something. I wasn't convinced that the
machine had the capabilities to record a video, but it did it seems.
Click above to watch/listen. Lyrics sheet available here.
I delivered this poem to Dr. Carolyn Otto's Discrete Math course (Math 314)
at UW-Eau Claire in the Spring of 2016. I was the substitute teacher
for Carolyn this particular day and on her lesson plan was an introduction
to graph theory via the well-known Bridges of Konigsberg problem.
Hence I was inspired to deliver to these students this poem that I wrote on
exactly this particular graph theory problem. Lyrics sheet available here.
Miscellaneous videos I made where I am not singing mathematics to
students
In this video, myself and my former students Katie Henschell (Quotes) and
Taylor Kriesel (Knit Mama) give a complete detailed proof of the celebrated
Snake Lemma. What is unique about this video is that students Quotes and
Knit Mama did this with me a few days after their graduation from UWEC in
May 2021. So instead of partying after graduation, they took a deep dive
into exploring the finer intricacies of this awesome result and delivering a
clear concise proof. YAY!!!
UWEC Math Club President Sara DeBrabander (2015-2016) and I made this video
to explain the mathematics behind the math club t-shirt which our math club
designed this year. It's a very cool t-shirt, but a little
understanding of the complex numbers and Euler's famous identity (melding e,
i , and pi) helps to better appreciate the t-shirt.
Yes, I got hooked by those videos of cats riding on top of robot Roomba
vacuums. So though I don't have a cat, I decided to buy this
vacuum. I must admit the unused vacuum of mine is now collecting more
dust on top of it then it does inside of it. Ha. This video
shows the vacuum in action for about 3 minutes, then I get bored of that and
decide to give an impromptu tour of my home and bicycle.
Teaching-Assistant for a course on Math for Elementary Education Majors
while I was a grad student at the University of Iowa
|
7 |
| ... sometimes teaching indoors on a nice day is such a drag -
that's when you need a blackboard on wheels!! |
This page was last modified on Oct. 31, 2023 (Jewish
Calendar: 16th of Cheshvan, 5784), by extra-planetary beings from
a galaxy near you. Go U Blugolds!!
Click
on
the bear below to go to the UWEC math faculty webpage:
PROCLAIMER: Click
on
"Proclaimer", and Tibetan Buddhist scholar Chogyam Trungpa Rinpoche
introduces this film up to the 4min marker. The meat of the film's
discussion begins at the 9min marker. Also when asked how/why I got
interested in and high
as a mountain on math, I usually answer with words like "outdoor raves
/ technivals
/ full moon gatherings / vacant
warehouse breakins
/ etc".
I found mathematics on many sunrises & moonsets in deserts, farms,
fields, and Pacific coast beaches--somewhere between minimal Detroit house,
deep house, tribal house, acid,
and hard acid
techno house (see map on House) at this link.
If you were one of the many beautiful souls who evolved with me
through our trancedelic bliss
& harmony on those mornings dancing to the sunrise in Northern
California, then you might recall the bass bins oozing Psychotropic
Hypnosis or perhaps Frankie Knuckles' Whistle
Song or perhaps this ANTHEM OF HOUSE MUSIC: Can
You Feel It? And of course the nights too under the stars, we
shared love and light while being transmutated and uplifted by housey beatz
like this, that,
and the other trancier thing
in our blessed Trance Frandisco, California, resonating this future
sound of SF.
Peace, Love, Unity, and Respect.
RESPECTS:
Love&LIGHT to Wicked,
Come-Unity, the Gathering, The End-Up's Housing Project, Your Sister's
House, Rave Called Sharon, the Sunset Crew, Friends and Family, Floppy's, Zippy Pronoia, Megatripolis, Autonomous
Mutant
Fest, Semi-Permanent Autonomous Zone (SPAZ),
Consortium of Collective Consciousness (CCC), and all the sound system
collectives! Lastly, respects to my Pleiadian mates and their message.
PRONOIA:
the sneaking suspicion that others are conspiring to help you (and you
them!). Symptoms of Pronoia include sudden attacks of optimism and outbreaks
of goodwill.
