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Home » Online Exhibits » A Brief History of Global Engagement at the University of Pennsylvania » Penn around the World: A History of Penn's Engagement with Specific Regions and Countries » Germany
A Brief History of Global Engagement at the University of Pennsylvania Penn around the World: A History of Penn's Engagement with Specific Regions and Countries Germany
German connections in the 18th and 19th centuries
John Christopher Kunze, c. 1790
In 1754, the teaching of the German language was begun at the Academy (Charity School) and at the College, no doubt due to the recommendation to the Academy Trustees by Rev. George Whitefield of a similar plan espoused by the pedagogy utilized at the University of Halle (now known as the Martin Luther University of Halle-Wittenberg). Due to the interest in the German language, after 1780, the year in which the Rev. Justus C. H. Helmuth and the Rev. Johan C. Kunze, Professors of German in the College, had become Trustees, a German School, conducted in the German language and led by the two aforementioned professors, was initiated and continued until 1792. Besides Helmuth and Kunze (pictured at right), other native German Trustees of the time included the Rev. Casparus D. Weyberg, who also received an honorary Master of Arts degree in 1780, the Rev. Ferdinand Steinmeyer, S.J. (also known as Ferdinand Farmer), and the Rev. Laurence Graessl.
As for students, excluding those who attended the German School from 1780 to 1792, the earliest German Penn graduate seems to be August T. Stamm (Berlin, Germany), who graduated from the Medical School in 1855. Two Germans also graduated in 1883 with the fifth class of the School of Dentistry.
World War I through World War II
Honorary degrees were given to many Germans in the first half of the twentieth century, but perhaps none more notable than the one awarded in 1905 to the German Kaiser Friedrich Wilhelm II. Another notable honorary degree, awarded in 1911, went to Johann Heinrich Von Bernstorff, the German Ambassador to the United States and Mexico from 1908 to 1917. Von Bernstorff is best known for his anti-American, pro-German sabotage attempts (some successful) conducted in New Jersey and elsewhere during World War I. These activities continued until President Woodrow Wilson broke off diplomatic relations with Germany, forcing Von Bernstorff to return to Germany in 1916. The Penn Trustees struck both Von Bernstorff and the Kaiser from the roll of honorary degree recipients January 14, 1918, making them the only two people in Penn history to have been struck from the honorary degree recipients roll.
In 1912, as Penn's interest in Germany seems to have faltered, the German-American Research Institute, led by Professor M. D. Learned ended. That same year aGerman Traveling Scholarship Fund was begun; it lasted for two academic years (to the beginning of the Great War). The Trustees then shifted remaining funds from this program in 1920 to the Jusserand Traveling Scholarship Fund, founded in 1912 by Joseph G. Rosengarten named for Jules Jusserand , the former French ambassador to the United States. Although by the 1920s, Penn was not immune to the anti-German sentiment felt throughout the world, some German graduates of the Penn School of Dentistry kept in touch with the University.
Elsa K. Nitzsche, Class of 1937, as a sophomore, 1935
In the 1930s, multiple arts performances on the Penn campus centered on German culture and sometimes starred German performers. The famous German contralto, Maragrete Olden, gave a recital to 1,700 people at Irvine Auditorium. Elsa K. Nitzsche (pictured below, left), Class of 1937 and the daughter of former University of Pennsylvania Recorder George Nitzsche, won a scholarship for the summer to study at Munich University (now known as Ludwig Maximilians University of Munich); she also performed the title role in "Maria Stuart" (by Schiller) staged by the German Club in Irvine Auditorium in 1935. Another German recital occurred in 1936 performed by the "distinguished German song interpreter" Hans Merz. The German Club throughout the 1930s performed a number of well-attended plays, including Franz Kafka's "Das Schloss" ("The Castle") (scene shown below, center) and Gotthold E. Lessing's play, "Minna von Barnhelm."
In 1939, twenty Swedish folk dancers (pictured below, right) performed at Penn; the group had been on a tour of the United States.
Swedish folk dancer, Mrs. Gunnel Henke, in ethnic costume, 1939
"Das Schloss" ("The Castle"), a play by Franz Kafka, a scene from the Penn performance, 1935
In 1940, an honorary degree was awarded to Reinhold Niebuhr, known for his philosophy on diplomacy and politics. Niebuhr was a Protestant theologian, and his studies of religious faith heavily influenced his writing; he became an influence for the idea of the modern "just war." Another honorary degree was awarded by Penn to Wolfgang Kohler, a German psychologist, in 1946. The University honored these men not as German political leaders like Von Bernstorff and the Kaiser, but as exceptional German intellectuals.
During World War II, the University of Pennsylvania offered its resources in support of the United States in a number of ways. As part of the University's war effort, Penn faculty trained students in the German language to enable them to better engage the enemy. In April 1942, it was reported that Penn was to offer a special summer course in "Nazi language." This term was not some sort of headline language used by the press; rather, it referred to a Penn course on specific German terminology developed by the Nazi Party and used in official party documents. The Army Specialized Training Program intensive course in German was led by Dr. Otto Springer, and this course was introduced to the civilian student population at Penn in late 1944. As already mentioned in the section of this exhibit on World War II, a Penn Committee on Post-War International Problems (similar committees were held across the nation at various institutions and the entire project was led by Harvard University) engaged issues such as how the world would deal with Post-War Germany. This is not to mention that the world would have to deal with two Germanys with two official ideologies: one communist and one Westernized and split three ways.
Post World War II
Cover of first Department of Germanic Languages newsletter, 1955
After World War II, many of the relations between Penn and Germany returned to as they had been in the 1930s – there was a large emphasis on cultural studies. In the second edition (1960) of the Department of Germanic Languages and Literature's Newsletter, a report explained that the Department wished to "effect a closer integration of German with related subjects such as teaching… and… a Pedagogical Seminar in collaboration with the School of Education." This Newsletter (cover pictured at right) had begun in 1955 as a bulletin for that department.
Medieval and Renaissance German songs were performed on December 10, 1964, by Karl Wolfram in College Hall. Films, representing examples of German expressionism from the years between the wars, were presented by the Philomathean Society during the week of April 23, 1967. "Die Bruecke," an internationally acclaimed "German theatrical ensemble" performed the following year and again in 1970 and 1972. In 1968 the Honorary Consul of the Netherlands, Dr. Jan W. S. Nordholt, lectured in Houston Hall on "Dumas and Adams: The First Two Representatives in Holland." "Classical Drama and Modern Theater" was the subject of a lecture by Dr. Wilhelm Emrich of the Freie Universität Berlin (Penn would send students to study here) at the Penn campus meeting of the Germanic Association in 1971. In 1972, a New Jersey digest reported on Penn's graduate studies in Netherlandic as well as other Scandinavian languages. A February 1973 exhibit at Van Pelt Library of reproductions of the work of famous Dutch painters from four centuries prompted visits to the Penn campus by W. S. J. Campagne, the consul general of the Netherlands in the United States and General L. Gleeson, the consul of the Netherlands in Philadelphia.
In 1976, the University allowed students to apply for the Junior Year in Munich program of Wayne State University, Detroit, Michigan, finally adding Germany to Penn's list of countries acceptable for study abroad. At the Faculty Club in January 1979, there was a celebration of the independence of Holland, commemorating the 400th anniversary of the signing of the Treaty of Utrecht. Modern films from the Federal Republic of Germany (FRG), were shown in the Annenberg Studio Theater in November, 1982, and in 1983, President Ronald Reagan and President Karl Carstens of the FRG were to appear on campus to be "at a conference commemorating the tricentennial anniversary of German settlement in America," but that visit did not occur.


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info@cognitivegroup.com (514) 861-1212
Banking Practice
Process Simplification
Deliver unmatched
Why Cognitive Group
We combine a deep understanding of human behavior and cognitive engineering to deliver unmatched simplicity of systems, solutions and processes.
The Cognitive Group (CG) was founded in 1995 by the Montreal Computer Research Institute (CRIM) to develop a cognitive engineering consulting practice. Cognitive engineering enables better design of decision systems.
Cognitive Group is a partner with Masetria and GDG. With 300 employees, Cognitive can access to an additional 1000 experts for specific client's needs.
Cognitive Group Head Office is in Montreal, Canada
History of Business Banker
In 2008, Cognitive Group was selected to review the loan origination process for the Royal Bank of Canada due to its unique expertise in cognitive engineering. For 10 years we covered all aspects of the loan origination process at RBC, including risk management, renewal automation, digital experience, sales, non negotiable securities, portfolio, document preparation and compliance.
Cognitive Group continued to apply its expertise at the National Bank of Canada, TD Insurance, Laurentian Bank, Desjardins, in China, Southeast Asia, Africa, the Caribbean and across the globe through technical assistance from the IFC (World Bank).
Cognitive, with the aid of partners and investors decided to invest and develop a complete end-to-end origination solution called "Business Banker". "Business Banker" incorporates our expertise and completely transforms the loan origination process. Instead of requiring six systems explained in manuals of over 1000 pages, our solution comes down to one system built from the ground up. It is entirely cloud-based and is extremely easy to use and maintain.
2015: Following RBC, Cognitive decided to invest three years to develop Business Banker, Businessbanker.io
2017: Deploy at BNDE
2018: Win contract with GBDK in Lisbon
2020: Deployed in six banks
2020: Work begin for implementation at DUCA
Francois Aubin Founder and CEO
Marie-Claude Prévost, Cognitive
Jacques Perron, Maestria
Gaétan Duchesne, GDG
Advisory committee:
Louis St Jacques, Archimed
Marc André Roy, R3D
Peter Gaston, Common Ground Solutions
1155 René-Lévesque Blvd West #2500
Montréal, QC H3B 2K4
info@cognitivegroup.com
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Marie-Claude Prevost
Marie-Claude is an Electrical Engineer with 20 years experience specialized in process optimisation, user experience (UX) and system design using a human factors approach. She has a unique ability to rapidly understand how people handle complex situations and design processes so that they are performed in the most efficient way.
She has consulting experience and repeat business with companies such as CAE, Hydro-Quebec, Gaz Metro, RBC, National Bank of Canada, Disnat, Desjardins, TD, Quebec National Library and Archives, Tele-Quebec. She worked on over than 50 projects involving Web, mobile, tablet, real time, SCADA, mission-5critical systems, client-server, IVR, telecom, portal, Intranet, search engines, policies & procedures, etc.
She is regularly invited as a speaker at international conferences and to give lectures at Ecole Polytechnique of the University of Montreal. She is actively involved in the Montreal community through various urban/social/art projects and as board member.
Francois Aubin
Francois has over 20 years of experience showing leading companies how to improve operational efficiency using human factors. Francois specializes in human factors and human computer interaction. He has been involved in over 100 IT projects and has a high level of expertise in business banking, mortgages processes. Francois's client list includes Hydro-Quebec, Bell Canada, Ericsson, Lassonde, Kronos, Desjardins and Royal Bank,National Bank, Amazon, PG&E, Shaw Group, V1Jet and American Pacific. Francois has a B.Sc in physics and a master's degree specialized in human factors from the Industrial Engineering Department of Ecole Polytechnique of Montreal. He has published numerous papers on 3D display, wearable computers, input devices, and task analysis. He frequently gives workshops on user experience at the University of Quebec and at École Polytechnique.
Peter Gaston
Peter has over 25 years of experience advising and leading companies to deliver business results at the intersection of business and technology. His experience spans business and IT consulting – with a high level of expertise in customer relationship management, product innovation including rapid prototyping, and enterprise architecture. Peter's client list includes HSBC, Royal Bank of Canada, Charles Schwab & Co., Fidelity Investments, Wells Fargo, ABN AMRO, CIGNA, Swiss Re, Farm Credit Canada, MassMutual, Fair Isaac, and Sprint/Nextel. Previously, Peter was a Vice President at Nextera Inc., a NASDAQ traded consulting company where he led the Boston office of over 150 staff and the customer relationship practice. Peter was a consultant with McKinsey & Co. in the area of IT Strategy and Product Innovation. Mr. Gaston has earned his S.B. and S.M. from M.I.T.
Sylvain Leith
Sylvain Leith is an expert in risk management and modeling. He has solid, wide-ranging experience in the field of credit risk management as well as in international banking regulations under the auspices of the Bank for International Settlements (BIS). Sylvain is an expert in risk-related governance and business intelligence modelling, specializing particularly in credit risk and economic capital. In that respect, he was vice-president in charge of a risk modelling and strategy department at the National Bank of Canada, acting as leader in implementing the provisions of the Basel II Accord, implementing a credit stress testing program, integrating advanced notation in risk assessment practices and adopting principles for risk data aggregation and risk notification. After leaving the National Bank, Sylvain worked as an expert consultant in risk management both in Canada and outside the country. He also implemented a credit risk policy manual and procedures on movable assets financing for two banks in China and liquidity impasse assessments for a bank in Haiti, respectively on behalf of the World Bank's International Finance Corporation (IFC) and the European Investment Bank. As a subject matter expert, Sylvain works with the advisory section of KPMG financial risk management.
Cognitive was founded in 1995 by the Computer Research Institute of Montreal. Our experts have advanced degrees in human factors, human computer interaction and interaction design. Our specialists are located in Montreal, Toronto, Hong Kong and San Francisco.
Decision support system causes a tragedy at Quebec Directorate of Youth Protection
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GWAAR
About SK
Change Food
On-Site Culinary Testing
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Policy and Procedure Development
Operational Systems
Marketing Campaigns and PR
Self Scout
What's In Your Spice Rack?
The spice rack. We all have one. Or in my case, what once was a spice cupboard replete with decade-old powders and amalgamated configurations of salt, bleached, then colored, then packaged and labeled as "fish seasoning" or "steak seasoning."
I've since purged most of these inherited dust vials, and today you will find nutmeg, sweet curry powder, cinnamon, saffron, lavender, Korean chili threads or Madagascar vanilla beans. More importantly, what you won't find is dried herbs, onion powder or garlic salt.
On a rainy Wednesday afternoon at the Frontier Airlines Center, I strolled through the annual Wisconsin Restaurant Expo in search of new and exciting products with which to enhance my menus and inspire my epicurean appetite.
As I surveyed a line-up of dried herbs and powdered "this and that," I asked the grandfatherly gentleman at the booth, "Are these all natural?" He responded proudly. "Yes, our spices are all natural." I then asked, "Are they irradiated?"
A big smile appeared on his face. I was talking his language now. As if to ease my worries about pathogen-infected seasonings, he stuck out his chin and delightfully proclaimed his line of herbs and spices to be, indeed, irradiated. I said thank you and moved on.
So, what's irradiation? Sounds like radiation, right? That's because it is. It's a process that is widely used to kill microorganisms in dried herbs and spices. Seems like a good idea doesn't it? "I don't want to get sick from my dried, crunchy, flavorless, swampy-green parsley, do I?" Of course you don't.
But aside from having little culinary value, garlic salt, onion powder and dried basil may not be particularly good for you either.
"But chef," you ask, "isn't there a thrilling nouvelle cuisine afoot called molecular gastronomy that is predicated on pulverizing commonplace ingredients and repurposing classic flavor combinations by dispelling their compositions at the molecular level?" Yes. While my personal culinary style may be a tad more wholesome, I respect the invention and innovation of this movement despite its price tag.
This, however, is not what I'm talking about. Certainly, you can find all sorts of intriguing re-constitutions on a plate such as sea salt foam or shitake mushroom powder. But, in virtually all instances, these inventions are realized in the restaurant kitchens themselves and are arrived at organically. It's sort of like the difference between someone being paid to create a large painted mural on the interstate overpass and someone just vandalizing it without sanction.
The reasoning behind irradiation is valid. The goal is plain. But, as with so many of the unpleasant truths we continue to learn about where our food comes from, irradiation is a tricky bird with all sorts of unknowable risks and a nuanced process that I hope will have you opting for fresh parsley, garlic cloves and those big white things that make you teary when you slice into them.
I could write an entire article about the flavor benefits of fresh ingredients versus dried. But, we'll just say, coming from a chef, that's self-evident. Instead, I'd like to pull back the curtain on this curious practice and the FDA-approved steps for making it "safe."
The intention of my culinary scribblings is partly to be entertaining, so I'm not about to put you to sleep by detailing the scientific "ins and outs" of irradiation. I will, however, touch on the main aspects of the process.
But first, let's understand that the expressed goal of this system is to eliminate potentially harmful pathogens from food to make it suitable for consumption. If there is a technology that can prevent us from contracting E.coli and salmonella from foods, we should probably want to support that (nevermind the question of how it got there to begin with).
Most simply put, food is sent on a conveyor through a concrete fortress (like the illustration above) and exposed to controlled amounts of ionizing radiation by way of a gun similar to a television tube.
From the UW-Madison website on food irradiation:
"When microbes present in the food are irradiated, the energy from the radiation breaks the bonds in the DNA molecules, causing defects in the genetic instructions. Unless this damage can be repaired, the organism will die or will be unable to reproduce."
This means that your dried herbs and spices are full of dead bugs. OK, maybe I'm exaggerating just a bit for effect but ... maybe not.
The FDA has approved irradiation of meat and poultry and allows its use for a variety of other foods including fresh fruits, vegetables and spices. Realize though, this is less to protect you and more to protect the food. It enables food to be grown, harvested and subsequently stored for much longer periods of time and with a decreased risk of spoilage.
Whole Foods Market asserts that "Not all food authorities support irradiation and even the strongest proponents acknowledge that irradiation is no match for bad sanitation and substandard practices. Irradiation can only help control the contamination once it occurs; it cannot prevent it."
Essentially, irradiation is a cop out. Its anticipated use can discourage strict adherence to good sanitary practices. Bill Freese, a science policy analyst with the Center for Food Safety in Washington
calls it a "pseudo fix."
"It's a way to try to come in and clean up problems that are created in the middle of the food production chain," says Freese. "I think it's clearly a disincentive to clean up the problems at the source."
While processors of irradiated food are subject to all existing regulations, there is no mandated regulatory oversight. In other words, they are not subject to regular plant inspections. This, in effect, necessitates the use of irradiation. You know that old saying about how you'd never eat a hot dog if you saw how it was made? Well, that may be true for your dried oregano as well.
Irradiation of fruits and vegetables and meat and poultry is not widely practiced yet. However, almost all wholesale dried herbs and spices are irradiated. Furthermore, the European Union has regulated that only dried herbs and spices may be subject to this process. That said, Europa, the European commission for food and feed safety, follows a far stricter code of food-flow from the source. Irradiation is used as a final "only-in-some-cases" measure.
There are many wonderful ethnic cuisines that are rooted in the use of dried spices. In Mexican or Indian cuisine, for example, where the path to deep rich flavors starts with what we know as "low and slow" cooking methods, these bright, wholesome spices only approach their brilliant apex through proper use and long, careful cooking.
For these wonderful dishes, dare you attempt them, there are un-irradiated spices available at places like Whole Foods Market, Trader Joe's or just your neighborhood spice shop.
If you're waiting for me to tell you that irradiation has been linked to diseases, birth defects or other maladies, I can't say that. But that's sort of the point – it's unknowable. Irradiation is another form of preventative maintenance born out of our industrial food production system.
The lack of long-term studies should be further reason for us to cease irradiating foods in order to attain more information about its life-long affects.
When the FDA says that this practice "does not change the nutritional value of foods and they don't become dangerous as a result of irradiation," what they're really saying is that no one else can prove otherwise. In essence, the lack of data is the very vehicle for its proliferation.
Now, I'm well aware of how exhausting it can be to keep up with all the latest atrocities of our food factories. There's always something to be appalled by. Maintaining a purely organic, natural and unfettered diet of as-nature-intended foods is virtually impossible in a world were organic food is not the standard but a commodity.
But, if the thought of the food you consume being exposed to gamma rays isn't enough of a cause for concern and your mind is adequately put at ease by the FDA's ardent claim that irradiation is "perfectly safe," I would only ask you this: why purchase anything that is dried or powdered when the fresh version of it is three aisles over?
Junk Food: The Story of Bad Relationship
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Sustainable Kitchens
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There's no magic. It's Tiani.
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Tiani Spirit is the backbone of Grapevine. Tiani already connects the world's largest health information exchanges.
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HomeMovies
Fast & Furious 7 casts Latin music superstar Romeo Santos!
By Divya Soni
Updated: 26th May 2014, 11:41 AM
The latest star to join the cast of Fast & Furious 7 is Latin-American singer Romeo Santos.
Romeo has signed up to play an unknown role in the new movie and the casting was announced on 20th May by Vin Diesel when he joined Romeo at his sold out gig in Los Angeles. Former lead singer of Bachata group Aventura, Romeo Santos has over 19 million fans on Facebook and 2 million followers on Twitter!
Also joining the cast of the eagerly awaited seventh movie are Jason Statham and Kurt Russell.
Fast and Furious 7 will be released on 10th April 2015 and will of course star Vin Diesel, Dwayne Johnson, Michelle Rodriguez, Jordana Brewster, Tyrese Gibson, Ludacris, Lucas Black and the late Paul Walker as the leading cast members.
Fast and Furious 7
Graham Norton to star in Will Ferrell's new movie
Giant Harry Potter wands are now lighting up London for J.K. Rowling's international children's charity Lumos
Harry Potter wands to light up London landmark ahead of Fantastic Beasts 2 release


The How to Play Banjo Handy Guide is a useful publication for every musician aiming to enter the world of banjo. This Alfred item is packed with useful info, featuring lessons on how play chords, the basic strums, many folk songs, banjo techniques, and more. One of the other factors worthy of mentioning is the simple fact that this product has a quite fair and affordable price, making it a must-have for any banjo aficionado out there.
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Just witnessed a white Ford Bantam unknown reg, black tinted windows with 2 bravo occupants drove into a lady on Cnr Spine road and Park Avenue Mitchell's plain. They clearly tried getting away as they ddnt even stop but sped off instead. A green goldish Kia with 4 male occupants followed the bakkie Kia had no number plates was clear that the two vehicles were in cahoots. I tried following them to obtain reg of Bantam bakkie as they drove recklessly over all the robots on Spine road direction khayelitsha. All occupants whore green reflector bibs it seemed as if they were involved in some criminal activity. If anyone has been targeted by these two vehicles above in the past 40min just know that they headed direction Khayelitsha on Spine road.


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\section{Introduction}
We consider state transfer with respect to the adjacency matrix of a graph (see \cite{bon}). All graphs presented are assumed to be finite, undirected and simple. Let $G$ be a graph on $n$ vertices with adjacency matrix $A$. The transition matrix associated to $G$ is defined by
\begin{eqnarray}\label{IE1}
H(t):=\exp{\left(-itA\right)},~\text{where}~t\in{\mathbb R}.
\end{eqnarray}
A quantum state in a quantum network based on $G$ is defined to be a unit vector in ${\mathbb C}^n$. The Equation \ref{IE1} governs the way a quantum state evolve over time. Let ${\bf e}_u$ denote the characteristic vector corresponding to the vertex $u$ of $G$. The graph $G$ is said to exhibit pretty good state transfer (PGST) between a pair of vertices $u$ and $v$ if there is a sequence $\left\lbrace t_k\right\rbrace$ of real numbers such that
\begin{eqnarray}\label{IE2}
\lim_{k\rightarrow\infty} H(t_k) {\bf e}_u=\gamma {\bf e}_v,~\text{where}~\gamma\in{\mathbb C}~\text{and}~|\gamma|=1.
\end{eqnarray}
If there is a sub-sequential limit $t_0$ of $\left\lbrace t_k\right\rbrace$ then Equation \ref{IE2} gives $H(t_0) {\bf e}_u=\gamma {\bf e}_v,$ in which case we say that there is perfect state transfer (PST) (see \cite{bose}) between $u$ and $v$ at time $t_0$. If $H(t_0) {\bf e}_u=\gamma {\bf e}_u$ for $t_0\neq 0$ then $G$ is said to be periodic at $u$ at time $t_0$. Moreover $G$ is said to be periodic if it is periodic at all vertices at the same time (in that case $H(t_0)=\gamma I$). Perfect state transfer is most desirable in this context, however, Godsil \cite{god2} has shown that there are very few graphs exhibiting PST. Consequently the study of PGST, introduced by Godsil in \cite{god1}, acquired considerable attention. In \cite{god4}, Godsil \emph{et al.} characterized all paths exhibiting PGST between the end vertices. Later, Coutinho \emph{et al.} \cite{cou3} further investigated and found a class paths having PGST between internal vertices. In \cite{pal4}, we found that a cycle on $n$ vertices exhibits PGST if and only if $n$ is a power of $2$. This was the first examples of non-integral circulant graphs admitting PGST. Apart from cycles, we have also found families of non-integral circulant graphs exhibiting PGST in \cite{pal0, pal4}. Several other ralavent results regarding state transfer can be found in \cite{mil4, fan, kirk, pal, pal1, pal2, pal3, saxe}. We extend some pre-existing results appearing in \cite{pal0, pal4} on circulant graphs admitting PGST and re-discover families of circulant graphs on $2^k$ vertices exhibiting PST.\par
We begin with a brief introduction to Kronecker approximation theorem on simultaneous approximation of numbers. We apply this result to establish some relations on the eigenvalues of a circulant graph in Lemma \ref{L3} and Theorem \ref{T4}.
\begin{theorem}[Kronecker approximation theorem]\cite{apo}
If $\alpha_1,\ldots,\alpha_l$ are arbitrary real numbers and if $1,\theta_1,\ldots, \theta_l$ are real, algebraic numbers linearly independent over ${\mathbb Q}$ then for $\epsilon>0$ there exist $q\in{\mathbb Z}$ and $p_1,\ldots,p_l\in{\mathbb Z}$ such that
\[\left|q\theta_j-p_j-\alpha_j\right|<\epsilon.\]
\end{theorem}
A Cayley graph over a finite abelian group $\left(\Gamma,+\right)$ with the connection set $S$, where $0\notin S\subseteq\Gamma$ and $\left\lbrace -s:s\in S\right\rbrace=S$, is denoted by $Cay\left(\Gamma,S\right)$. The elements of $\Gamma$ are the vertices of the graph, where two vertices $a,b\in\Gamma$ are adjacent if and only if $a-b\in S$. If $\Gamma={\mathbb Z}_n$ then the Cayley graph is called circulant graph. In particular, a Cayley graph over ${\mathbb Z}_n$ with $S=\left\lbrace1,n-1\right\rbrace$ is called a cycle which is denoted by $C_n$. The eigenvalues of $C_n$ are given by
\begin{eqnarray}\label{IE3}
\lambda_l=2\cos{\left(\frac{2l\pi}{n}\right)},\;0\leq l\leq n-1,
\end{eqnarray}
and the associated eigenvectors are ${\mathbf v}_l=\left[1,\omega_n^l,\ldots,\omega_n^{l(n-1)}\right]^T$, where $\omega_n=\exp{\left(\frac{2\pi i}{n}\right)}$ is the primitive $n$-th root of unity.\par
Let the graphs $G$ and $H$ be defined on a common vertex set $V$, and the respective edge sets $E(G)$ and $E(H)$. The edge union of $G$ and $H$, denoted $G\cup H$, is a graph with vertex set $V$ and edge set $E(G)\cup E(H)$. The following result enables us to realise the transition matrix of a Cayley graph as a product of transition matrices of its sub-graphs.
\begin{prop}\cite{pal1}\label{IP1}
Let $\Gamma$ be a finite abelian group and consider two disjoint and symmetric subsets $S,T\subset\Gamma$. Suppose the transition matrices of $Cay(\Gamma, S)$ and $ Cay(\Gamma, T)$ are $H_{S}(t)$ and $H_{T}(t)$, respectively. Then $Cay(\Gamma, S\cup T)$ has the transition matrix $H_{S}(t)H_{T}(t).$
\end{prop}
A graph is called integral if all its eigenvalues are integers. Let $n,d\in{\mathbb N}$ and $d$ be a proper divisor of $n$. We denote $S_n(d)=\left\lbrace x\in{\mathbb Z}_n: gcd(x,n)=d\right\rbrace.$ If $D$ is a set containing proper divisors of $n$ then define $S_n(D)=\bigcup\limits_{d\in D} S_n(d).$ The set $S_n(D)$ is called a gcd-set of ${\mathbb Z}_n$. The following theorem due to W. So characterizes all integral circulant graphs.
\begin{theorem}\cite{so}\label{so}
A circulant graph $Cay\left({\mathbb Z}_n,S\right)$ is integral if and only if $S$ is a gcd-set.
\end{theorem}
Hence, for a non-integral circulant graph $Cay\left({\mathbb Z}_n,S\right)$, there must exist at least one such $d$ so that $S\cap S_n(d)$ is a non-empty proper subset of $S_n(d)$.\par
Now we include the proof the following proposition which was mentioned in \cite{fan} without a proof. The result gives a relation between PST and PGST in periodic graphs.
\begin{prop}\label{IP2}
If a graph is periodic, then it admits perfect state transfer if and only if it admits pretty good state transfer.
\end{prop}
\begin{proof}
It is easy to observe that if a graph $G$ has PST then it must also have PGST. On the other hand, suppose that a graph is periodic at $\tau$ and therefore for a vertex $u$ of $G$ there exists $\gamma\in{\mathbb C}$ with $|\gamma|=1$ so that $H(\tau){\bf e}_u=\gamma{\bf e}_u$. For $u,v\in V(G)$, the $uv$-th entry of $H(t)$ is a continuous function of real numbers. Also observe that, for $t\in{\mathbb R}$,
\begin{eqnarray*}
\left|{\bf e}_v^T H(t+\tau){\bf e}_u\right| = \left|{\bf e}_v^T H(t)H(\tau){\bf e}_u\right|
&=& \left|{\bf e}_v^T H(t)\left(\gamma{\bf e}_u\right)\right|,\text{ as }H(\tau){\bf e}_u=\gamma{\bf e}_u\\
&=& \left|{\bf e}_v^T H(t){\bf e}_u\right|.
\end{eqnarray*}
Therefore, the modulus of the $uv$-th entry of $H(t)$ is a periodic continuous function and so its image is a compact set. Since $H(t)$ is unitary, the modulus of each entry of $H(t)$ is bounded by $1$ for all $t\in{\mathbb R}$. Finally, if it satisfies Equation \ref{IE2} then by extreme value theorem there exists $\tau_0\in{\mathbb R}$ so that $|{\bf e}_v^T H(\tau_0){\bf e}_u|=1,$ \emph{i.e,} the graph $G$ has PST between $u$ and $v$. Hence the result follows.
\end{proof}
In the following section we introduce some previously known results on PGST in circulant graphs and after that we include the main results with some examples for convenience.

\section{Pretty Good State Transfer on Circulant Graphs}
We begin the discussion with few relevant results on circulant graphs exhibiting PGST appearing in \cite{pal0,pal4}. It is well known that if a graph $G$ admits PGST between two vertices $u$ and $v$ then all automorphisms of $G$ that fix $u$ must also fix $v.$ This leads to the conclusion which we present as follows.
\begin{lem}\label{L1}\cite{pal4}
If pretty good state transfer occurs in a circulant graph $Cay\left({\mathbb Z}_n,S\right)$ then $n$ is even and it occurs only between the pair of vertices $u$ and $u+\frac{n}{2}$, where $u,u+\frac{n}{2}\in{\mathbb Z}_n$.
\end{lem}
A circulant graph $G$ being vertex transitive, for any pair of vertices $u$, $v$ in $G$ there exists an automorphism mapping $u$ to $v$. Let $A$ be the adjacency matrix of $G$. Note that the transition matrix $H(t)$ of $G$ can be realised as a polynomial in $A$. If $P$ is the matrix of an automorphism of $G$ then $P$ commutes with $A$ as well as $H(t)$. If $G$ exhibits PGST between two vertices $u$ and $v$ then Equation \ref{IE2} infers that
\[\lim_{k\rightarrow\infty} H(t_k) \left(Pe_u\right)=\gamma \left(Pe_v\right).\]
by Lemma \ref{L1}, it is enough to find PGST in a circulant graph between the pair of vertices $0$ and $\frac{n}{2}$. Let the spectral decomposition of the adjacency matrix of $C_n$ be $A=\sum\limits_{l=0}^{n-1}\lambda_lE_l,$ where $E_l=\frac{1}{n}{\mathbf v}_l{\mathbf v}_l^*$ and $\lambda_l,~{\mathbf v}_l$ is as mentioned in Equation \ref{IE3}. Therefore, from Equation \ref{IE1}, the transition matrix of $C_n$ is evaluated as
\[H(t)=\exp{\left(-itA\right)}=\sum\limits_{l=0}^{n-1}\exp{\left(-i\lambda_l t\right)E_l}.\]
Note that $\left(0,\frac{n}{2}\right)$-th entry of $E_l$ is $\frac{1}{n}\omega_n^{-\frac{nl}{2}}$. Hence $(0,\frac{n}{2})$-th entry of $H(t)$ is given by
\begin{eqnarray}\label{E1}
H(t)_{0,\frac{n}{2}}=\frac{1}{n}\sum\limits_{l=0}^{n-1}\exp{\left(-i\lambda_l t\right)}\cdot\omega_{n}^{-\frac{nl}{2}}=\frac{1}{n}\sum\limits_{l=0}^{n-1}\exp{\left[-i\left(\lambda_l t+l\pi\right)\right]}.
\end{eqnarray}
It is shown in \cite{saxe} that if a cycle admits PST then it must be integral and this leads to the conclusion that only $C_4$ exhibits PST (see \cite{mil4}). We further investigated and in \cite{pal4}, we have found a characterization of all cycles admitting PGST.
\begin{theorem}\label{T1}\cite{pal4}
A cycle $C_n$ admits pretty good state transfer if and only if $n=2^k$ for $k\geq 2.$
\end{theorem}
 In \cite{wal1}, it appears that the eigenvectors of a Cayley graph over an abelian group are independent of the connection set. Therefore the set of eigenvectors of two Cayley graphs defined over the same abelian group can be chosen to be equal. Hence we have the following result appearing in \cite{pal1}.
\begin{prop}\cite{pal1}\label{IP3}
If $S_1$ and $S_2$ are symmetric subsets of an abelian group $\Gamma$ then adjacency matrices of the Cayley garphs $Cay(\Gamma, S_1 )$ and $Cay(\Gamma, S_2 )$ commute.
\end{prop}
Let the eigenvalues of a circulant graph $Cay({\mathbb Z}_n, S)$ be $\theta_l,$ for $l=0,1,\ldots,n-1$. If $H_S(t)$ is the transition matrix of $Cay({\mathbb Z}_n, S)$ then, by Proposition \ref{IP3}, we obtain
\begin{eqnarray}
H_S(t)_{0,\frac{n}{2}} &=&\frac{1}{n}\sum\limits_{l=0}^{n-1}\exp{\left[-i\left(\theta_l t+l\pi\right)\right]},\label{E2}\\
H_S(t)_{0,0}&=&\frac{1}{n}\sum\limits_{l=0}^{n-1}\exp{\left(-i\theta_l t\right)}.\label{E2a}
\end{eqnarray}
A graph $G$ is said to be almost periodic if there is a sequence $\left\lbrace t_k\right\rbrace$ of non-zero real numbers and a complex number $\gamma$ of unit modulus such that $\lim\limits_{k\rightarrow\infty} H(t_k) =\gamma I,$ where $I$ is the identity matrix of appropriate order. Since the circulant graphs are vertex transitive, we observe that a circulant graph is almost periodic if and only if $\lim\limits_{k\rightarrow\infty} H(t_k) {\bf e}_0=\gamma {\bf e}_0.$ This implies that if a cycle admits PGST then it is necessarily almost periodic. Next we find a connection between the time sequences of two cycles one of which exhibits PGST and the other one is almost periodic.

\begin{lem}\label{L2}\cite{pal0}
Let $k,k'\in {\mathbb N}$ and $2\leq k'<k$. If the cycle $C_{2^k}$ admits pretty good state transfer with respect to a sequence $\left\lbrace t_m\right\rbrace$ then $C_{2^{k'}}$ is almost periodic with respect to a sub-sequence of $\left\lbrace t_m\right\rbrace$.
\end{lem}

Apart from the cycles, using Lemma \ref{L2}, Theorem \ref{T1} has been extended in \cite{pal0} to obtain more circulant graphs on $2^k$ vertices exhibiting PGST. Also, there we find some circulant graphs which are almost periodic. We include the result for convenience.

\begin{theorem}\label{T2}\cite{pal0}
Let $k\in{\mathbb N}$ and $n=2^k$. Also let $Cay\left({\mathbb Z}_n,S\right)$ be a non-integral circulant graph. Let $d$ be the least among all the divisors of $n$ so that $S\cap S_n(d)$ is a non-empty proper subset of $S_n(d)$. If $\left|S\cap S_n(d)\right|\equiv 2\pmod{4}$ then $Cay\left({\mathbb Z}_n,S\right)$ admits pretty good state transfer with respect to a sequence in $2\pi{\mathbb Z}$. Moreover, if $\left|S\cap S_n(d)\right|\equiv 0\pmod{4}$ then $Cay\left({\mathbb Z}_n,S\right)$ is almost periodic with respect to a sequence in $2\pi{\mathbb Z}$.
\end{theorem}
Note that Theorem \ref{T2} fails to decide whether $Cay\left({\mathbb Z}_n,S\right)$ admits PGST whenever $\left|S\cap S_n(d)\right|\equiv 0\pmod{4}$. Computational results suggest that few such graphs exhibits PGST. Consider the following example.
\begin{ex}\label{Ex1}
Let $n=16,~S=\{1,2,3,4,12,13,14,15\}$ and consider the circulant graph $G=Cay\left({\mathbb Z}_n,S\right)$. Observe that $S\cap S_n(1)=\{1,3,13,15\}\subsetneq S_n(1).$ Thus, when considering PGST, Theorem \ref{T2} is inconclusive in this case. However, calculating the $\left(0,\frac{n}{2}\right)$-th entry of the transition matrix of $G$, we obtain the following outcome presented in Figure \ref{f:p1} with the help of GNU Octave \cite{eat}. Later we shall show that $G$ exhibits PGST indeed.
\begin{figure}[h!]
\begin{subfigure}{0.4\textwidth}
\begin{tikzpicture}[thin, scale=0.7, every node/.style={scale=0.8}]
\grCirculant{16}{1,15,3,13,2,14,4,12}
\end{tikzpicture}
\end{subfigure}
\hfill
\begin{subfigure}{0.6\textwidth}
\includegraphics[width=25em]{pex1.png}
\end{subfigure}
\caption{\small The $\left(0,\frac{n}{2}\right)$-th entry of transition matrix of $Cay\left({\mathbb Z}_n,S\right)$ at $[2*7500*\pi,~2*8000*\pi]\cap 2\pi{\mathbb Z}$.}
\label{f:p1}
\end{figure}
\end{ex}

Now we shall prove the following result, revisiting the proof of Lemma[6] in \cite{pal4}, on the eigenvalues of circulant graphs.
\begin{lem}\label{L3}
Let $k,k'\in{\mathbb N}$ be such that $n=2^k$ and $d=2^{k'}$ with $\frac{n}{d}\geq 8$. Let $\lambda_l,$ where $l=0,1,\ldots,n-1$, be the eigenvalues of $C_n$. For $\delta>0$ there exists $t\in 2\pi{\mathbb Z}$ so that for each $l$ there exists an integer $l'$ such that
\begin{eqnarray*}
\begin{cases}
\left|\left(\lambda_l t+a\pi\right)- 2l'\pi\right|<\frac{\delta}{n}, & \text{ if $l=2^{k'}\cdot a$ and $a$ is odd.}\\
\left|\lambda_l t- 2l'\pi\right|<\frac{\delta}{n}, & \text{ otherwise.}
\end{cases}
\end{eqnarray*}
\end{lem}
\begin{proof}
We first show that the distinct positive eigenvalues of $C_n$ are linearly independent over ${\mathbb Q}$. The eigenvalues of $C_n$ can be realized as $\lambda_l=\omega_n^l+\omega_n^{-l}.$ Note that for $ 1\leq l\leq 2^{k-2}-1$, the relation $\lambda_l=-\lambda_{\frac{n}{2}-l}=-\lambda_{\frac{n}{2}+l}=\lambda_{n-l}$ holds, and the remaining eigenvalues $\lambda_0,\lambda_{2^{k-2}},\lambda_{2^{k-1}}$ and $\lambda_{3\cdot2^{k-2}}$ are integers. It is well known that the minimal polynomial of $\omega_n$ over ${\mathbb Q}$ has degree $\phi(n)$, where $\phi$ is Euler's phi-function (see \cite{esc}). The distinct positive eigenvalues are $\lambda_l$ where $0\leq l<\frac{n}{4}.$ If the distinct positive eigenvalues are dependent over ${\mathbb Q}$ then $\omega_n$ will be a root of a polynomial of degree at most $m=2^{k-1}-1$ as $\omega_n^{-1}=-\omega_n^{m}$. But since $2^{k-1}-1<2^{k-1}= \phi(2^k),$ we conclude that the distinct positive eigenvalues are linearly independent over ${\mathbb Q}$. Now, for $ 1\leq l\leq 2^{k-2}-1$, assume that
\begin{eqnarray*}
\alpha_l=\begin{cases} \frac{1}{2}, & \text{ if $l=2^{k'}\cdot a$ and $a$ is odd.}\\
0, & \text{ otherwise.}
\end{cases}
\end{eqnarray*}
By Kronecker approximation theorem, for $\delta>0$ there exist $q,m_1,\ldots,m_{2^{k-2}-1}\in{\mathbb Z}$ such that for $l=1,\ldots, 2^{k-2}-1$
\begin{eqnarray}\label{L3E1}
\left|q\lambda_l-m_l-\alpha_l\right|<\frac{\delta}{2n\pi},\; \emph{i.e,}\;\left|2q\pi\lambda_l-2m_l\pi-2\alpha_l\pi\right|<\frac{\delta}{n}.
\end{eqnarray}
Since $\lambda_0,\lambda_{2^{k-2}},\lambda_{2^{k-1}}$ and $\lambda_{3\cdot2^{k-2}}$ are integers, considering $t=2q\pi$, we observe that for each $l=0,\;2^{k-2},\;2^{k-1},\;3\cdot2^{k-2}$ there exists an integer $l'$ such that
\[\left|\lambda_l t- 2l'\pi\right|<\frac{\delta}{n}.\]
Also for each $l=1,\ldots, 2^{k-2}-1$, considering $t=2q\pi$ and using Equation \ref{L3E1}, we find
\begin{eqnarray*}
l'=\begin{cases} \frac{2m_l+a+1}{2}, & \text{ if $l=2^{k'}\cdot a$ and $a$ is odd.}\\
m_l, & \text{ otherwise.}
\end{cases}
\end{eqnarray*}
such that
\begin{eqnarray*}
\begin{cases}
\left|\left(\lambda_l t+a\pi\right)- 2l'\pi\right|<\frac{\delta}{n}, & \text{ if $l=2^{k'}\cdot a$ and $a$ is odd.}\\
\left|\lambda_l t- 2l'\pi\right|<\frac{\delta}{n}, & \text{ otherwise.}
\end{cases}
\end{eqnarray*}
Since $\lambda_l=-\lambda_{\frac{n}{2}-l}=-\lambda_{\frac{n}{2}+l}=\lambda_{n-l}$ holds for $ 1\leq l\leq 2^{k-2}-1$, considering $t=2q\pi$, we conclude that for each $l=0,\ldots, n-1,$ there is an integer $l'$ such that
\begin{eqnarray*}
\begin{cases}
\left|\left(\lambda_l t+a\pi\right)- 2l'\pi\right|<\frac{\delta}{n}, & \text{ if $l=2^{k'}\cdot a$ and $a$ is odd.}\\
\left|\lambda_l t- 2l'\pi\right|<\frac{\delta}{n}, & \text{ otherwise.}
\end{cases}
\end{eqnarray*}
This completes the proof.
\end{proof}
It is well known that if a circulant graph exhibits PGST then it is almost periodic. In Theorem \ref{T2}, we find that if $\left|S\cap S_n(d)\right|\equiv 0\pmod{4}$ then the graph $Cay\left({\mathbb Z}_n,S\right)$ is almost periodic, and thus presumably many such graphs exhibits PGST. We present a subclass of those circulant graphs admitting PGST extending Theorem \ref{T2}.
\begin{theorem}\label{T3}
Let $k\in{\mathbb N}$ and $n=2^k$. Also let $Cay\left({\mathbb Z}_n,S\right)$ be a non-integral circulant graph. Let $d$ be the least among all the divisors of $n$ so that $S\cap S_n(d)\neq \emptyset$ and $S\cap S_n(d)\subsetneq S_n(d)$ with $\left|S\cap S_n(d)\right|\equiv 2\pmod{4}$. Then $Cay\left({\mathbb Z}_n,S\right)$ admits pretty good state transfer with respect to a sequence in $2\pi{\mathbb Z}$.
\end{theorem}

\begin{proof}
First we shall prove that there exists a sequence in $2\pi{\mathbb Z}$ with respect to which $Cay\left({\mathbb Z}_n,S\cap S_n(d)\right)$ exhibits PGST and $Cay\left({\mathbb Z}_n,S\setminus S_n(d)\right)$ is almost periodic. Let us suppose $d=2^{k'}.$ Since $S\cap S_n(d)\neq \emptyset$ and $S\cap S_n(d)\subsetneq S_n(d)$, the set $S_n(d)$ must have atleast $4$ elements. Therefore we must have $\phi\left(\frac{n}{d}\right)\geq 4,$ i.e, $\frac{n}{d}\geq 8$.\\
The exponential function $\exp{\left(-ix\right)}$ is uniform continuous. Therefore, for $\epsilon>0$, there exists $\delta>0$ such that $x_1,x_2\in{\mathbb R}$ and $|x_1-x_2|<\delta$ implies $|\exp{\left(-ix_1\right)}-\exp{\left(-ix_2\right)}|<\epsilon.$ Now we proceed using the following steps.\\
\textbf{Step 1:} If $\theta_l$ denotes the eigenvalues of $Cay\left({\mathbb Z}_n,S\cap S_n(d)\right)$ then
\[\theta_l=\frac{1}{2}\sum\limits_{s\in S\cap S_n(d)}\lambda_{ls}.\]
Since $\left|S\cap S_n(d)\right|\equiv 2\pmod{4}$, using Lemma \ref{L3} and triangle inequality, we obtain that for $\delta>0$ there exists $t\in 2\pi{\mathbb Z}$ such that for each $l$ there exists an integer $l'$ such that
\begin{eqnarray*}
\left|\left(\theta_l t+l\pi\right)- 2l'\pi\right|<\delta
\end{eqnarray*}
Hence there exists $t=2\pi{\mathbb Z}$ such that $\left|\exp{\left[-i\left(\theta_l t+l\pi\right)\right]}- 1\right|<\epsilon.$ Finally, by Equation \ref{E2}, if $H_{S\cap S_n(d)}(t)$ is the transition matrix of $Cay\left({\mathbb Z}_n,S\cap S_n(d)\right)$ then
\begin{eqnarray*}
\left|\left[H_{S\cap S_n(d)}(t)\right]_{0,\frac{n}{2}}-1\right|=\frac{1}{n}\left|\sum\limits_{l=0}^{n-1}\left(\exp{\left[-i\left(\theta_l t+l\pi\right)\right]}-1\right)\right|<\epsilon.
\end{eqnarray*}
This leads to the conclusion that $Cay\left({\mathbb Z}_n,S\cap S_n(d)\right)$ admits PGST with respect to a sequence $\left\lbrace t_m\right\rbrace$ (say) in $2\pi{\mathbb Z}$. Hence
\begin{eqnarray}\label{T3E1}
\lim_{m\rightarrow\infty} H_{S\cap S_n(d)}\left(t_m\right) {\bf e}_0={\bf e}_{\frac{n}{2}}
\end{eqnarray}
\textbf{Step 2:} If $d'$ be a divisor of $n$ such that $d'<d$ and $S\cap S_n(d')\neq \emptyset$ then we have $\left|S\cap S_n(d')\right|\equiv 0\pmod{4}.$ If $\eta_l$ denotes the eigenvalues of $Cay\left({\mathbb Z}_n,S\cap S_n(d')\right)$ then
\[\eta_l=\frac{1}{2}\sum\limits_{s\in S\cap S_n(d')}\lambda_{ls}.\]
Since $\left|S\cap S_n(d)\right|\equiv 0\pmod{4}$, using Lemma \ref{L3} and triangle inequality, we can carefully choose the same $t\in 2\pi{\mathbb Z}$ as in Step 1 so that for each $l$ there exists an integer $l'$ such that
\begin{eqnarray*}
\left|\eta_l t- 2l'\pi\right|<\delta
\end{eqnarray*}
Hence there exists $t=2\pi{\mathbb Z}$ such that $\left|\exp{\left[-i\eta_l t\right]}- 1\right|<\epsilon.$ Finally, if $H_{S\cap S_n(d')}(t)$ is the transition matrix of $Cay\left({\mathbb Z}_n,S\cap S_n(d')\right)$ then Equation \ref{E2a} implies
\[\left|\left[H_{S\cap S_n(d')}(t)\right]_{0,0}-1\right|=\frac{1}{n}\left|\sum\limits_{l=0}^{n-1}\left(\exp{\left[-i\eta_l t\right]}-1\right)\right|<\epsilon.\]
This leads to the conclusion that $Cay\left({\mathbb Z}_n,S\cap S_n(d')\right),$ where $d'<d$, is almost periodic with respect to the same sequence $\left\lbrace t_m\right\rbrace$ as obtained in Step 1.  Therefore we have
\begin{eqnarray}\label{T3E2}
\lim\limits_{m\rightarrow\infty} H_{S\cap S_n(d')}\left(t_m\right)=I.
\end{eqnarray}
\textbf{Step 3:} If $d''$ be a divisor of $n$ such that $d''>d$ and $S\cap S_n(d'')\neq \emptyset$ then for $s\in S\cap S_n(d'')$ the size of each disjoint cycles appearing in $Cay\left({\mathbb Z}_n,\left\lbrace s,n-s\right\rbrace\right)$ is strictly less than the size of the cycle $C_{n/d}$. In Step 1, in particular, if we consider $S\cap S_n(d)=\{d,-d\}$ then $Cay\left({\mathbb Z}_n,S\cap S_n(d)\right)$, which is disjoint union of $d$ copies of the cycle $C_{n/d}$, admits PGST with respect to $\left\lbrace t_m\right\rbrace$. Hence the cycle $C_{n/d}$, in particular, admits PGST with respect to $\left\lbrace t_m\right\rbrace$. Therefore, by Lemma \ref{L2}, we have a subsequence of $\left\lbrace t_m\right\rbrace$ with respect to which each component of $Cay\left({\mathbb Z}_n,\left\lbrace s,n-s\right\rbrace\right),~s\in S\cap S_n(d''),$ is almost periodic. Hence $Cay\left({\mathbb Z}_n,\left\lbrace s,n-s\right\rbrace\right)$ is also almost periodic. Since $s\in S\cap S_n(d'')$ is finite, we can appropriately choose a subsequence $\left\lbrace t_{m_r}\right\rbrace$ of $\left\lbrace t_m\right\rbrace$ such that for all $s\in S\cap S_n(d'')$, the graph $Cay\left({\mathbb Z}_n,\left\lbrace s,n-s\right\rbrace\right)$ is almost periodic with respect to $\left\lbrace t_{m_r}\right\rbrace.$ Hence, by Proposition \ref{IP1}, the graph $Cay\left({\mathbb Z}_n,S\cap S_n(d'')\right)$ is almost periodic with respect to $\left\lbrace t_{m_r}\right\rbrace.$ Therefore, if $H_{S\cap S_n(d'')}(t)$ is the transition matrix of $Cay\left({\mathbb Z}_n,S\cap S_n(d'')\right)$ then
\begin{eqnarray}\label{T3E3}
\lim\limits_{m\rightarrow\infty} H_{S\cap S_n(d'')}\left(t_{m_r}\right)=I.
\end{eqnarray}
Notice that $S_n(d_1)\cap S_n(d_2)=\emptyset$ whenever $d_1\neq d_2$. Therefore, by Proposition \ref{IP1} and using Equation \ref{T3E2} and Equation \ref{T3E3}, we obtain
\begin{eqnarray}\label{T3E4}
H_{S}\left(t_{m_r}\right)
&=& \left[\prod\limits_{d'|n,~d'<d} H_{S\cap S_n(d')}\left(t_{m_r}\right)\right]\cdot\left[\prod\limits_{d''|n,~d<d''} H_{S\cap S_n(d'')}\left(t_{m_r}\right)\right]\cdot H_{S\cap S_n(d)}\left(t_{m_r}\right) \nonumber \\
&=& H_{S\cap S_n(d')}\left(t_{m_r}\right).
\end{eqnarray}
Finally, by Equation \ref{T3E1} and Equation \ref{T3E4}, we conclude that $Cay\left({\mathbb Z}_n,S\right)$ admits PGST with respect to a sequence in $2\pi{\mathbb Z}$.
\end{proof}
Recall that, in Example \ref{Ex1}, the graph $Cay\left({\mathbb Z}_n,S\right)$ satisfying all the conditions of Theorem \ref{T3} as $S\cap S_n(2)=\{2,4\}$. Hence $Cay\left({\mathbb Z}_n,S\right)$ admits PGST with respect to a sequence in $2\pi{\mathbb Z}$. Using Theorem \ref{T3}, we can find many such graphs exhibiting PGST. However notice that if, for all $d$ with $S\cap S_n(d)\subsetneq S_n(d)$, we have $\left|S\cap S_n(d)\right|\equiv 0\pmod{4}$ then Theorem \ref{T3} does not apply. Consider the following example.
\begin{ex}\label{Ex2}
Let $n=16,~S=\{1,3,4,12,13,15\}$ and consider the circulant graph $G=Cay\left({\mathbb Z}_n,S\right)$. Observe that $S\cap S_n(1)=\{1,3,13,15\}\subsetneq S_n(1).$ Obtaining the $\left(0,\frac{n}{2}\right)$-th entry of the transition matrix of $G$, we observe the following Figure \ref{f:p2} in GNU Octave \cite{eat}. The next result suggests that $G$ exhibits PGST.
\begin{figure}[h!]
\begin{subfigure}{0.4\textwidth}
\begin{tikzpicture}[thin, scale=0.7, every node/.style={scale=0.8}]
\grCirculant{16}{1,3,4,12,13,15}
\end{tikzpicture}
\end{subfigure}
\hfill
\begin{subfigure}{0.6\textwidth}
\includegraphics[width=25em]{pex2.png}
\end{subfigure}
\caption{\small The $\left(0,\frac{n}{2}\right)$-th entry of transition matrix of $Cay\left({\mathbb Z}_n,S\right)$ at $[0,500\pi]\cap \left(2{\mathbb Z}+1\right)\frac{\pi}{2}$.}
\label{f:p2}
\end{figure}
\end{ex}
In such cases mentioned above, we show that if either $\frac{n}{2}$ or $\frac{n}{4}$ but not both is in $S$ then $Cay\left({\mathbb Z}_n,S\right)$ admits PGST. Consider the following theorem.
\begin{theorem}\label{T4}
Let $k\in{\mathbb N}$, $n=2^k$ and $Cay\left({\mathbb Z}_n,S\right)$ be a circulant graph with $\left|\left\lbrace\frac{n}{2},~\frac{n}{4}\right\rbrace \cap S\right|\equiv 1\pmod{2}$. If each divisor $d~\left(\neq \frac{n}{2},\frac{n}{4}\right)$ of $n$ satisfy $\left|S\cap S_n(d)\right|\equiv 0\pmod{4}$ then $Cay\left({\mathbb Z}_n,S\right)$ admits pretty good state transfer with respect to a sequence in $\left(2{\mathbb Z}+1\right)\frac{\pi}{2}$.
\end{theorem}

\begin{proof}
We have established in Lemma \ref{L3} that the distinct positive eigenvalues of $C_n$ are linearly independent over ${\mathbb Q}$. For $ 1\leq l\leq 2^{k-2}-1$, let us choose $\alpha_l=-\frac{\lambda_l}{2}$. By Kronecker approximation theorem, for $\delta>0$ there exist $q,m_1,\ldots,m_{2^{k-2}-1}\in{\mathbb Z}$ such that for $l=1,\ldots, 2^{k-2}-1$
\begin{eqnarray}\label{T4E1}
\left|q\lambda_l-m_l-\alpha_l\right|<\frac{\delta}{n\pi},\; \emph{i.e,}\;\left|\left((2q+1)\frac{\pi}{2}\right)\lambda_l-m_l\pi\right|<\frac{\delta}{n}.
\end{eqnarray}
Since $\lambda_0,\lambda_{2^{k-2}},\lambda_{2^{k-1}}$ and $\lambda_{3\cdot2^{k-2}}$ are even integers, considering $t=\left(2q+1\right)\frac{\pi}{2}$, we observe that for each $l=0,\;2^{k-2},\;2^{k-1},\;3\cdot2^{k-2}$ there exists an integer $l'$ such that
\[\left|\lambda_l t- l'\pi\right|<\frac{\delta}{n}.\]
Since $\lambda_l=-\lambda_{\frac{n}{2}-l}=-\lambda_{\frac{n}{2}+l}=\lambda_{n-l}$ holds for $ 1\leq l\leq 2^{k-2}-1$, considering $t=\left(2q+1\right)\frac{\pi}{2}$ and using Equation \ref{T4E1}, we conclude that for each $l=0,\ldots, n-1,$ there is an integer $l'$ such that
\begin{eqnarray}\label{T4E2}
\left|\lambda_l t- l'\pi\right|<\frac{\delta}{n}.
\end{eqnarray}
Consider $S'=S\setminus\left\lbrace\frac{n}{4},\frac{3n}{4},\frac{n}{2}\right\rbrace$. If $\theta_l$ denotes the eigenvalues of $Cay\left({\mathbb Z}_n,S'\right)$ then
\[\theta_l=\frac{1}{2}\sum\limits_{s\in S'}\lambda_{ls}.\]
Since each divisor $d~\left(\neq \frac{n}{2},\frac{n}{4}\right)$ of $n$ satisfy $\left|S\cap S_n(d)\right|\equiv 0\pmod{4}$, using Equation \ref{T4E2} and triangle inequality, we obtain that for $\delta>0$ there exists $t\in\left(2{\mathbb Z}+1\right)\frac{\pi}{2}$ so that for each $l$ there exists an integer $l'$ such that
\begin{eqnarray*}
\left|\theta_l t- 2l'\pi\right|<\delta
\end{eqnarray*}
Since the exponential function is uniform continuous, for $\epsilon>0$ there exists $t\in\left(2{\mathbb Z}+1\right)\frac{\pi}{2}$ such that $\left|\exp{\left[-i\theta_l t\right]}- 1\right|<\epsilon.$ Finally, if $H_{S'}(t)$ is the transition matrix of $Cay\left({\mathbb Z}_n,S'\right)$ then by Equation \ref{E2a}, we obtain
\begin{eqnarray*}
\left|\left[H_{S'
}(t)\right]_{0,0}-1\right|=\frac{1}{n}\left|\sum\limits_{l=0}^{n-1}\left(\exp{\left[-i\theta_l t\right]}-1\right)\right|<\epsilon.
\end{eqnarray*}
This leads to the conclusion that $Cay\left({\mathbb Z}_n,S'\right)$ almost periodic with respect to a sequence $\left\lbrace t_m\right\rbrace$ (say) in $\left(2{\mathbb Z}+1\right)\frac{\pi}{2}$. Hence
\begin{eqnarray}\label{T4E3}
\lim_{m\rightarrow\infty} H_{S'}\left(t_m\right)=I.
\end{eqnarray}
By Proposition \ref{IP1} and using Equation \ref{T4E3}, we find that if $H_{S}(t)$ and $H_{S\setminus S'}(t)$ are the transition matrices of $Cay\left({\mathbb Z}_n,S\right)$ and $Cay\left({\mathbb Z}_n,S\setminus S'\right)$, respectively, then
\begin{eqnarray}\label{T4E4}
\lim_{m\rightarrow\infty} H_{S}\left(t_m\right)=\lim_{m\rightarrow\infty} H_{S\setminus S'}\left(t_m\right).
\end{eqnarray}
Notice that $Cay\left({\mathbb Z}_n,\left\lbrace\frac{n}{2}\right\rbrace\right)$ is the disjoint union of path on two vertices (denoted by $P_2$) and $Cay\left({\mathbb Z}_n,\left\lbrace\frac{n}{4},\frac{3n}{4}\right\rbrace\right)$ is the disjoint union of $C_4$. It is well known that both $P_2$ and $C_4$ admits perfect state transfer at $\frac{\pi}{2}$ and are periodic at $\pi$. Hence, by Equation \ref{T4E4}, we conclude that $Cay\left({\mathbb Z}_n,S\right)$ exhibits PGST with respect to a sequence in $t\in\left(2{\mathbb Z}+1\right)\frac{\pi}{2}$.
\end{proof}
In Theorem \ref{T4}, if $Cay\left({\mathbb Z}_n,S\right)$ is integral then it is certainly periodic at $2\pi.$ Therefore, by Proposition \ref{IP2}, if $Cay\left({\mathbb Z}_n,S\right)$ exhibits PGST then $Cay\left({\mathbb Z}_n,S\right)$ also exhibits PST. A complete characterization of circulant graphs exhibiting perfect state transfer is appearing in \cite{mil4}. Notice that Theorem \ref{T4} in fact reveals all integral circulant graphs on $2^k$ vertices exhibiting PST. We present the scenario with an example.
\begin{ex}\label{Ex3}
Consider $S=\{1,2,3,5,6,7\}$ and $G=Cay\left({\mathbb Z}_8,S\right).$ By Theorem \ref{so}, the graph $G$ is integral and hence it is periodic at $2\pi$ (see the spectral decomposition of the transition matrix). Also applying Theorem \ref{T4}, we find that $G$ exhibits PGST. Finally, by Proposition \ref{IP2}, we conclude that $G$ exhibits perfect state transfer.
\end{ex}

Note that if a circulant graph exhibits PGST then it is almost periodic. In Theorem \ref{T2}, we find that if $\left|S\cap S_n(d)\right|\equiv 0\pmod{4}$ then $Cay\left({\mathbb Z}_n,S\right)$ is almost periodic. It is therefore tempting to presume that all such graph exhibits PGST, however, that is not the case. We demonstrate this with the following example.
\begin{ex}\label{Ex3}
Consider $S=\{1,7,9,15\}$ and $G=Cay\left({\mathbb Z}_{n},S\right),$ with $n=16$. Let the eigenvalues of $G$ be $\theta_l,$ where $l=0,1,\ldots,n-1$. Note here that $\theta_1=\theta_4=0$. If $G$ admits PGST then we have a sequence of real numbers $\left\lbrace t_k\right\rbrace$ and a complex number $\gamma$ with $|\gamma|=1$ such that (using Equation \ref{E2})
\[\lim_{k\rightarrow\infty}\sum\limits_{l=0}^{n-1}\exp{\left[-i\theta_l t_k+l\pi\right]}=n\gamma.\]
Since the unit circle is compact, we have a subsequence $\left\lbrace t'_k\right\rbrace$ of $\left\lbrace t_k\right\rbrace$ such that
\[\lim_{k\rightarrow\infty}\exp{[-i\theta_l t'_k+l\pi]}=\gamma,\text{ for }0\leq l\leq n-1.\]
\end{ex}
Now $\theta_1=0$ gives $\gamma=-1$ and $\theta_4=0$ gives $\gamma=1$, which is absurd. Hence the graph $G$ does not exhibit PGST.
\section*{Acknowledgements}
We are deeply grateful to Dr. Himadri Nayak, IIIT Bhagalpur, India, for the help with GNU Octave programming. Also we are indebted to the anonymous reviewer(s) for carefully assessing the manuscript.

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LEVY, FELIX ALEXANDER (1884–1963), U.S. Reform rabbi and scholar. Levy was born in New York, son of parents of Alsatian origin. He was ordained at Hebrew Union College (1907). From 1908 until his retirement in 1955, Rabbi Levy served Emmanuel Congregation, Chicago. He influenced his colleagues in modifying the attitude of Reform Judaism to halakhah and the nature of Jewish identity. These changes were embodied, inter alia, in the 1937 Columbus Platform adopted by the Central Conference of American Rabbis under Levy's leadership as president (1935–37). After his retirement, Rabbi Levy served as editor of Judaism and as dean of the Academy for Higher Jewish Learning in New York. A selection of his papers and sermons appeared in His Own Torah (ed. S.D. Temkin, 1969).
S.D. Temkin (ed.), His Own Torah (1969), 3–43.
"Levy, Felix Alexander." Encyclopaedia Judaica. . Encyclopedia.com. 24 Apr. 2019 <https://www.encyclopedia.com>.


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text:
Artist from Blue Apple wins exhibition
Several artists from Unisson's Blue Apple Art Studios at Terrey Hills entered Willoughby City Council's 'Connect, Collaborate, Celebrate' art exhibition.
The exhibition, now in it's fourth year, showcases the artistic works of people living with disability, people experiencing social isolation, or people at risk of marginalisation.
Celebrating International Day of People with Disability 2022, the competition and exhibition provides a platform for artists of all abilities to promote themselves and expand their creative talent.
This year, Unisson client Alison Hall won first prize for an acrylic on paper painting.
Alison is a long-term client of Unisson, who has been attending our art studios for many years. It is wonderful to see the positive impact of our studios in the community, and it's great to see Alison's contributions recognised in such a public way.
The winning entry, titled 'Flitter Flutter', took around 4 weeks to complete.
Alison says, "I like birds and colour. I see lots of birds when I go to Blue Apple (art studios in the bush at Terrey Hills). I also like to watch them from my window at home."
Part of Alison's prize includes a solo art exhibition in 2023 at the newly refurbished Doughterty Centre in Chatswood. She will be busy in the coming months working on her collection.
We extend our congratulations to Alison, and to all the other artists who participated in the exhibition.
'Flitter Flutter' by Alison Hall
Two of Unisson's supported independent living clients from the Central Coast region, Mike and Siobhan, recently entered the 'Splash of Colour Art &...
Art at the Library
Blue Apple Art Studios has welcomed local community support from both Warringah Mall and Forrestville libraries over the past four years. Both...
All the Colours of the Rainbow
Sydney artist and Unisson Supported Independent Living client, Victoria Atkinson, 50, was recently named as a finalist in the 2021 Archibald Prize out...


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text:
Q: How to remove tabs on multiple rows at once This is not a programming question but an inconvenience in the android-studio editor.
If you have an unwanted tab before all your lines, how can you remove them all at once? Now I have to manually go over 50 lines to remove all the tabs to make my code look clean.
If you want to add multiple tabs at once you just select all your code and press the tab button. So I'm looking for the reverse of that.

A: If I understood you right, you want to beautify the code itself. Fortunately, you don't have to do that manually - at all.
There's a keybinding for it, which may vary by your OS and which layout you use by default. Go to file -> settings -> Keymap and search for auto-indent. Here's what I get on Windows 10 with the default keymap:

Again, which you have may depend on OS (I'm assuming that mainly applies to Mac though) and your keymap, but you can automatically indent your code as per language standards using Ctrl+Alt+I.
Note that this mainly does indentation. If you chose to golf code and want to ungolf it, this will not work. At least it doesn't for Java.
However: This only works with code files the IDE or plugins support. This won't work for i.e. a .txt file out of the box.

If I misunderstood you and you only want to remove tabs without doing the auto-indent, there's at least two other options.
The first option is using multiple cursors. You can add an additional cursor with shift+alt+a mouse click where you want the cursor, or holding the mouse wheel and moving the cursor with the mouse wheel held down. There might be other methods as well, but those are the two I know of.
Once you have multiple cursors, delete the tabs just like normal. But be careful! Doing so might delete the entire line itself. If it does, you can do 1 tab/n units per indent level to the left, and press delete instead.
There's (AFAIK) no limit to how many cursors you can have at once, but you can theoretically do it with 50 lines at once if you want to. But general advice - don't add more cursors than you can see at once. These do run in parallel and it's easy to lose track if you're not careful, and you might end up deleting stuff you didn't want to delete.

And finally, the regex solution:
Note: Be careful with this. If you use it incorrectly, you might get unwanted results.
If you only want to do this in a limited area, highlight it first. Then CTRL+R and you'll be presented with the regular replace menu. Make sure Regex and In Selection are selected.
A base regex to go off is ^([\s]{2,4}|\t). Explanation just for reference:
^ - At the start of the line
(
    \s{4}   - Match 4 spaces
    |\t     - Or a tab character
)

Replace with nothing and click "replace all" (or just use the regular "replace" button if you want to double-check before you do anything). This will replace one occurrence of 4 spaces or a single tab character. If you use indentation that isn't based on 4, change the number.
This is only useable and useful if you've found yourself with incorrect indentation that's the same across all the relevant lines - it will not fix indentation mistakes and/or inconsistencies such as 3-space indentation when you want 4, or random indetation for the same block. Use the first or alternatively second method for that instead.

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