Applied mathematics

Euclidean lattices have attracted considerable research interest as they can be used to construct efficient cryptographic schemes that are believed to be quantum-resistant. The NTRU problem, introduced by J. Hoffstein, J. Pipher, and J. H. Silverman in 1996 [16], serves as an important average-case computational problem in lattice-based cryptography. Following their pioneer work, the NTRU assumption and its variants have been used widely in modern cryptographic constructions such as encryption, signature, etc.
Let Rq = Zq[x]/ (xn + 1) be a quotient polynomial ring. The standard NTRU problem asks to recover short polynomials f, g E Rq such that h - g/ f (mod q), given a public key h and the promise that such elements exist. In practice, the degree n is often a power of two. As a generalization of NTRU, the Module-NTRU problems were introduced by Cheon, Kim, Kim, and Son (IACR ePrint 2019/1468), and Chuengsatiansup, Prest, Stehle, Wallet, and Xagawa (ASIACCS '20).
In this thesis, we presented two post-quantum Digital Signature Schemes based on the Module-NTRU problem and its variants.

The following dissertation investigates algebraic frames and their spaces of minimal prime elements with respect to the Hull-Kernel topology and Inverse topology. Much work by other authors has been done in obtaining internal characterizations in frame-theoretic terms for when these spaces satisfy certain topological properties, but most of what is done is under the auspices of the finite intersection property. In the first half of this dissertation, we shall add to the literature more characterizations in this context, and in the second half we will study general algebraic frames and investigate which, if any, of the known theorems generalize to algebraic frames not necessarily with the FIP.
Throughout this investigative journey, we have found that certain ideals and filters of algebraic frames play a pivotal role in determining internal characterizations of the algebraic frames for when interesting topological properties occur in its space of minimal prime elements. In this dissertation, we investigate completely prime filters and compactly generated filters on algebraic frames. We introduce a new concept of subcompact elements and subcompactly generated filters. One of our main results is that the inverse topology on the space of minimal prime elements is compact if and only if every maximal subcompactly generated filter is completely prime. Furthermore, when the space of minimal prime elements is compact, then each minimal prime has what we are calling the compact absoluteness property.

The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four body problem admits certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection, or homoclinic web. In this thesis, the planar restricted four body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the stable/unstable manifolds of these spatial periodic orbits and look for intersections between these manifolds near the fundamental planar homoclinics. In this way, we are able to continue all of the basic planar homoclinic motions into the spatial problem as homoclinics for appropriate vertical Lyapunov orbits which, by the Smale Tangle theorem, suggest the existence of chaotic motions in the spatial problem. While the saddle-focus equilibrium solutions in the planar problems occur only at a discrete set of energy levels, the cycle-to-cycle homoclinics in the spatial problem are robust with respect to small changes in energy. The method uses high order Fourier-Taylor and Chebyshev series approximations in conjunction with the parameterization method, a general functional analytic framework for invariant manifolds. Tools that admit a natural notion of a-posteriori error analysis. Finally, we develop and implement a validation algorithm which we later use to obtain Theorems confirming the existence of homoclinic dynamics. This approach, known as the Radii polynomial, is a contraction mapping argument which can be applied to both the parameterized manifold and the Chebyshev arcs. When the Theorem applies, it guarantees the existence of a true solution near the approximation and it provides an upper bound on the C0 norm of the truncation error.

How one behaves after interacting with a friend may not be the same as before
the interaction began What factors a ect the formation of social interactions
between people and, once formed, how do social interactions leave lasting changes on
individual behavior? In this dissertation, a thorough review and conceptual synthesis
is provided Major features of coordination dynamics are demonstrated with
examples from both the intrapersonal and interpersonal coordination literature that
are interpreted via a conceptual scheme, the causal loops of coordination dynamics
An empirical, behavioral study of interpersonal coordination was conducted to
determine which spontaneous patterns of coordination formed and whether a remnant
of the interaction ensued ("social memory") To assess social memory in dyads, the
behavior preceding and following episodes of interaction was compared In the
experiment, pairs of people sat facing one another and made continuous flexion-extension finger movements while a window acted as a shutter to control
whether partners saw each other's movements Thus, vision ("social contact") allowed
spontaneous information exchange between partners through observation Each trial consisted of three successive intervals lasting twenty seconds: without social contact
("me and you"), with social contact ("us"), and again without ("me and you")
During social contact, a variety of patterns was observed ranging from phase coupling
to transient or absent collective behavior Individuals also entered and exited social
coordination differently In support of social memory, compared to before social
contact, after contact ended participants tended to remain near each other's
movement frequency Furthermore, the greater the stability of coupling, the more
similar the partners' post-interactional frequencies were Proposing that the
persistence of behavior in the absence of information exchange was the result of prior
frequency adaptation, a mathematical model of human movement was implemented
with Haken-Kelso-Bunz oscillators that reproduced the experimental findings, even
individual dyadic patterns Parametric manipulations revealed multiple routes to
persistence of behavior via the interplay of adaptation and other HKB model
parameters The experimental results, the model, and their interpretation form the
basis of a proposal for future research and possible therapeutic applications

This report describes the development of a low cost open source semiautonomous
robotic car and a way to communicate with it. It is a continuation of
prior research done by other students at FAU and published in recent ASEE
conferences.
The objective of this project was the development of a new robotic
platform with improved precision over the original, while still keeping the cost
down. It was developed with the aim to allow a hands-on approach to the
teaching of mathematics topics that are taught in the K-12 syllabus.
Improved robustness and reliability of the robotic platform for visually
solving math problems was achieved using a combination of PID loops to keep
track of distance and rotation. The precision was increased by changing the
position of the encoders to the shafts of each motor. A mobile application was developed to allow the student to draw the
geometric shapes on the screen before the car draws them. The mobile
application consists of two parts, the canvas that the user uses to draw the figure
and the configure section that lets the user change the parameters of the
controller.
Results show that the robot can draw standard geometric and complex
geometric shapes. It has high precision and sufficient accuracy, the accuracy can
be improved with some mechanical adjustments. During testing a Pythagorean
triangle was drawn to show visually the key mathematics concept.
The eventual goal of this project will be a K-12 class room study to obtain
the feedback of the teachers and students on the feasibility of using a robotic car
to teach math. Subsequent to that necessary changes will be made to
manufacture a unit that is easy to assemble by the teacher.