Hindu-Arabic Numerical System

The decimal number system, an integral part of daily life, traces its origins back to 6th-century India. Characterized by the digits zero through nine, revolutionized numerical computation and record-keeping, setting the stage for advancements in mathematics, science, and commerce. Despite its apparent simplicity and utility to the contemporary observer, the widespread adoption of this system across the globe was a gradual process that spanned over a millennium.

The genesis of the modern digits can be linked to the Brahmi numerals from the 3rd century, with the concept of zero emerging concurrently as a placeholder within the Babylonian base-60 counting system. This incorporation of zero into the Hindu-Arabic numerals was a significant leap forward, enabling a positional numeral system akin to the one in use today. Unlike the Roman numerals used in the Western world, which complicated written calculations and hindered efficient computation, the Hindu-Arabic system offered a streamlined approach by introducing a zero and limiting the numerals to nine, with each position signifying an increasing power of ten.

The spread of the Hindu-Arabic numeral system was propelled by the Islamic conquests of the 7th century, eventually being known as the Hindu-Arabic number system. However, its adoption in Europe faced considerable resistance, with Roman numerals predominating until the 12th century. The turning point came with Leonardo of Pisa, better known as Fibonacci, who, through his travels in Arab lands, was exposed to this efficient numeral system. His seminal work, Liber Abaci in 1202, introduced the Hindu-Arabic number system to Europe, but also demonstrated its practical application in commerce, thereby catalyzing a shift in European mathematical practices and commercial operations.

The transition from Roman to Hindu-Arabic numerals in Europe was marked by a protracted debate between the algorists, who advocated for the Hindu-Arabic system, and the abacists, who favored the traditional Roman numerals and counting boards. The dispute eventually subsided in the 16th century, leading to the relegation of Roman numerals to specific contexts and the ascendance of the Hindu-Arabic system as the preeminent numerical standard.

Diophantine Equations

Contributions of Diophantus of Alexandria hold a distinguished place. His seminal work, Arithmetica, unveiled in the 3rd century CE, is a key in the study of number theory, particularly in the realm of integers. This ancient text, encapsulating 130 equations, laid the foundation for what are now known as Diophantine equations—equations constrained to integer solutions.

Diophantine equations are a specialized subset of polynomial equations, which in their essence, are expressions consisting of one or more algebraic terms with at least one unknown variable, typically denoted by $x$. Diophantus’s pioneering approach, which sought integer solutions to these equations, earned him the title “the father of algebra,” a testament to his profound influence on the field, despite the modern algebraic notations and concepts that were to emerge centuries later.

A quintessential Diophantine problem might pose a scenario such as: “A father’s age is one less than twice the age of his son. If the digits of the father’s age are reversed to represent the son’s age, what are their ages?” The resolution to such problems often involves an initial trial-and-error methodology, culminating in a singular solution—in this instance, a father aged 73 and a son aged 37.

Despite being occasionally perceived as mere mathematical puzzles, Diophantine equations have posed significant challenges, with some remaining unsolved for millennia. A notable moment in the history of these equations occurred in 1637 when Pierre de Fermat, while engaging with Diophantine puzzles, conjectured an equation that appeared unsolvable: “If integer $n$ is greater than 2, then there are no three integers where $x^n + y^n = z^n$.” This marginal note in his copy of Arithmetica, devoid of a proof, laid the groundwork for what would become known as Fermat’s Last Theorem—a conundrum that would not be resolved until 1994.

Diophantus’s legacy extends beyond his mathematical works, encapsulated poetically in the riddle inscribed on his tombstone. This riddle, a polynomial equation in itself, challenges the reader to deduce Diophantus’s age based on key life events. The solution, ingeniously woven into the fabric of his life’s timeline, reveals his age to be 84 years.

The Arithmetica of Diophantus and the ensuing Diophantine equations exemplify the enduring nature of mathematical inquiry. These equations serve as a bridge between the ancient and modern mathematical worlds, illustrating the timeless quest for knowledge and the inherent beauty found in the pursuit of solutions to seemingly insurmountable problems. Diophantus’s work, transcending the centuries, continues to inspire mathematicians, affirming the profound impact of ancient scholarship on contemporary mathematical thought and exploration.

An Improbable Life by D.C. Wilcox, and the $k-\omega$ Model

I just finished reading the autobiography of D. C. Wilcox. He wrote a number of books that were published through his own company. One of the most popular is on fluid dynamics. A less known book is on turbulence modeling. He was famous for a particular two-equation turbulence model in the form of $k-\omega$. It is mostly known as the Wilcox $k-\omega$ model today. Dr. Wilcox had an interesting life, and one publication is called “An Improbable Life,” which is an autobiography. In the autobiography, he discusses his childhood, father, mother, growth as a child through an arrest. While incarcerated he studied and was helped by volunteers. He was admitted to MIT through American standardized testing. He then went on to obtain a Ph.D. at Caltech. He was very focused later in life, and tried to give back to the community that helped him. He was an outspoken conservative and a huge fan of Ayn Rand, and in particular Atlas Shrugged. So many people in his generation were influenced by the book. My only dissapointment with the autobiography was that he did not talk at all about his interest and development / motivation for his famous turbulence model. Otherwise, the little hard to find book is a short read, which was motivated to help other young men such as himself. Here are some favorite quotes:

Just over nine years passed from the day I woke up in a six foot by nine foot prison cell determined to reclaim my life to the day Caltech President Harold Brown handed me my Doctorate. I had taken a journey that required a great deal of hard work, some good fortune and the assistance of some wonderful people. Among those wonderful people, the gracious and generous Jean Kane Foulke du Pont stood out not only as a benefactor but, eventually, as a dear friend.

I visited with Mrs. du Pont just after I graduated from MIT in 1966 to thank her for all she had done for me. I offered to pay her back with installments over time so other scholars could get the kind of education she had made possible for me. “Oh, heavens no,” she said, “we wouldn’t know how to handle it with the IRS.”

She went on to tell me that I could repay her by making sure my children went to their college of choice. It wasn’t much of a repayment I thought, because Barbara and I couldn’t imagine doing otherwise.

I sent her a letter in 1986 to tell her that I had just fulfilled part of my promise to make sure my children received the college education they wanted. My daughter Kinley would be graduating from college almost twenty years to the day from when I had graduated from MIT. Sadly, a reply came back telling me that Mrs. du Pont had passed away.

D. C. Wilcox, “An Improbable Life,” Published by DCW Industries, Inc., 2007, ISBN 10: 1928729509

In late June of 1966, Barbara and I moved to California where a job awaited me with Douglas Aircraft in Long Beach. This was yet another one of my childhood dreams come true. I had never forgotten my wonderful days in California with Aunt Isabel and her son Warren. This time I was coming to California to stay.

After working for a year at Douglas Aircraft, I was accepted for graduate study at the California Institute of Technology where I met my third great teacher, Dr. Philip Saffman. In addition to being a wonderful teacher and $\mathrm{PhD}$ thesis adviser, he told me that he felt a truly dedicated student should pursue his studies like a monk in a monastery. For students who did that, he added, six years from high school to $\mathrm{PhD}$ should be the norm.

In June of 1970, after just three years of study under the guidance of this brilliant mathematician/scientist, I graduated with a PhD in Aeronautics. I had accomplished the second of my highschool goals by earning a PhD with just six years of college.

While I was at Caltech, my son Robert Sabatino Wilcox named after his two grandfathers – was born. The year was 1969. Dad would have been tickled to know that his grandson was born in the year that Neil Armstrong became the first man to walk on the moon.

After a brief time working for various Southern California aerospace companies, I founded my own company, which I named DCW Industries. The company came into existence on July 19 , 1973. Since I was twenty-nine years old, I had accomplished my third goal. Initially focused on aerospace research, the company prospered and I have published more than seventy scientific reports and journal articles in some of the aerospace industry’s most prestigious journals. The company now specializes in book publishing, and I have written several college-level textbooks that are used in universities all over the world.

Mom and I wrote each other from time to time until she passed away in 1977. In one letter she told me something that proved to be one of the nicest things she ever did for me. She told me that I had a relative who was a professor at UCLA. His name was Bill Meecham and he was my Aunt Mabel’s son. Since Mabel was Dad’s sister, we were first cousins.

I contacted Bill and discovered that we had more than our bloodline in common – we were both working in the same field! We became friends and he helped me obtain a part-time teaching job at UCLA in 1981. I have been a fixture in the Mechanical and Aerospace Engineering Department ever since.

When Aunt Mabel died in 1987, Bill showed me some genealogical information that was among her belongings. I noticed that Dad had a brother named Arthur who had died in 1947 when I was three years old. At the age of 43 , I had just discovered that I had an uncle I was completely unaware of. Of even greater significance, this information revealed a great deal about my family roots dating back through ten generations in America.

D. C. Wilcox, “An Improbable Life,” Published by DCW Industries, Inc., 2007, ISBN 10: 1928729509

On Algorithm

An algorithm, a concept rooted in 9th-century Arabic scholarship, is a methodical procedure for problem resolution, eliminating the need for trial-and-error. This term, reflecting centuries of intellectual endeavor, denotes the evolution from Euclid’s ancient formulations to Al-Khwarizmi’s systematic methods and Ada Lovelace’s 19th-century innovations, highlighting algorithms’ integral role in computational development.

Salome

It is my yearly tradition to listen to the opera Salome by Richard Strauss on Valentine’s Day. To celebrate, I treated the department staff to donuts. Working on my research while listening to the opera can be a bit distracting, but it makes for a perfect Valentine’s Day tradition. I first saw the opera while working at NASA Langley Research Center in Norfolk, Virginia, as part of the Virginia Opera Guild.

Remembering Professor Olga Alexandrovna Ladyzhenskaya Twenty Years On

It has been approximately twenty years since Professor Olga Alexandrovna Ladyzhenskaya passed away.

An eminent mathematician and member of several Academies of Science, passed away in January 2004. Her distinguished career was marked by significant contributions to partial differential equations, particularly the Navier–Stokes equations and nonlinear elliptic and parabolic equations.

Born in 1922 in Kologriv, Russia, Ladyzhenskaya’s passion for natural sciences was ignited by her father, a high school mathematics teacher. Tragically, he was executed by the NKVD in 1937, which led to Ladyzhenskaya being barred from Leningrad State University. Despite this, she persisted and graduated with honors from high school in 1939. She eventually studied at Pokrovskii Pedagogical Institute in Leningrad and later, at Moscow State University, graduating in 1947. That same year, she married A. A. Kiselev and joined Leningrad State University for graduate studies under the guidance of S. L. Sobolev.

Ladyzhenskaya’s academic journey continued as she became a postgraduate student under V. I. Smirnov and later led a seminar on mathematical physics and boundary-value problems. In 1953, she defended her habilitation dissertation at Moscow State University and in 1954, joined the Steklov Mathematical Institute in Leningrad. There, she collaborated with notable mathematicians and contributed significantly to the field of mathematical physics.

Her research focused primarily on two areas: the Navier–Stokes equations, and nonlinear elliptic and parabolic equations. In 1951, she proved a fundamental inequality for elliptic operators and explored the convergence of the Fourier method for hyperbolic equations. Her work in the late 1950s on the multiplicative inequality led to the proof of the existence of global unique solutions for two-dimensional Navier–Stokes systems. She also collaborated with A. A. Kisielev to demonstrate global existence in three-dimensional cases for small initial data and external forces. Ladyzhenskaya’s contributions extended to proving global existence of stationary and regular axially symmetric solutions to Navier–Stokes equations.

Her investigations into the regularity and uniqueness of weak Hopf solutions were groundbreaking. She showed that if a weak solution belongs to a specific function space, it is unique and regular. These achievements, along with her other work, were detailed in her monographs on Navier–Stokes equations and in collaboration with other mathematicians on nonlinear elliptic and parabolic equations.

AIAA SciTech 2024 – Parametric Study of the Hypersonic Near-Field and Sonic Boom from Waveriders using a Fully-Parabolized Approach

Citation: King, C. B., Shepard, C. T., and Miller, S. A. E., “Parametric Study of the Hypersonic Near-Field and Sonic Boom from Waveriders using a Fully-Parabolized Approach,” AIAA SciTech, Orlando, FL, Jan. 8-12, AIAA 2024-2106, 2024. DOI: 10.2514/6.2024-2106

Abstract: A parametric study is performed to understand the relationship between volume displace- ment, lift, near-field signature, and sonic boom overpressure for variable wedge angle power-law waveriders. The width of a parametric waverider is varied for freestream Mach numbers from 5 to 7. Both near-field and sonic boom predictions are made with a fully parabolized approach. The Upwind Parabolized Navier-Stokes solver is used to spatially march the hypersonic flow- field in the streamwise direction. The waveform parameter method is used to propagate the hypersonic near-field to the ground from a fixed altitude of 15.85 km. We find the magnitude of the SPL varies with frequency as −19.7 log 𝒇 . There is a positive quasi-linear relationship between near-field and sonic boom overpressures with volume displacement. For 𝑴∞ = 7, a 150% volume increase yields 92.5% and 60.9% rises in near-field and sonic boom overpressures, respectively. The effect of losses due to thermo-viscous effects and atmospheric absorption are quantified. We show that for a waverider of volume ∀ = 4970 cm3 at 𝑴∞ = 7, these losses, predicted by PCBoom using modules PCBurg and enhanced Burgers’ decrease maximum overpressure by 41.6% and 39.5% relative to WPM, respectively.

Priestess of Delphi (Oracle or Pythia)

A twenty year dream came true this December, 2023, as I traveled to Adelaide, Australia to view John Collier’s Priestess of Delphi (1891), the Oracle, or Pythia. I was able to view the painting for two days.

I am not afraid to say that the experience was overwhelming, and I definately had tears in my eyes. I am not a religious person, but it was what I believe people experience when they have religious inspiration or revelation.

People traveled all over the world to consult the Oracle. How is my journey different?

Colors from my camera, and the gallery skylight cast a small glare, but helped illuminate canvas and brushtrokes. Overtime, the sun came and set, casting new reflections and colors, letting me see the painting in new ways. I took 500 high quality photographs of the painting with different light, angles, details, and far-away.

Artist: https://en.wikipedia.org/wiki/John_Collier_(painter)

Painting: https://www.agsa.sa.gov.au/collection-publications/collection/works/priestess-of-delphi/25000/

Thank you to the people and museum in AUS SA at the AGSA for the experience.