We study the local well-posedness in the Sobolev space H^s for the modified Korteweg-de Vries (mKdV) equation on the real line. Kenig-Ponce-Vega \cite{KPV2} and Christ-Colliander-Tao established that the data-to-solution map fails to be uniformly con...
We consider generalizations of Schuetzenberger's promotion operator on the set L of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on L. By assigning weights to the edges of the graph in two different way...
Rowmotion is a certain well-studied bijective operator on the distributive lattice $J(P)$ of order ideals of a finite poset $P$. We introduce the rowmotion Markov chain ${\bf M}_{J(P)}$ by assigning a probability $p_x$ to each $x\in P$ and using thes...
Natural environments pose significant challenges for autonomous robot navigation, particularly due to their unstructured and ever-changing nature. Hiking trails, with their dynamic conditions influenced by weather, vegetation, and human traffic, repr...
We give the optimal conditions for the existence of a club consisting of former regular over an inaccessible and a measurable. The foricing construction based on iteration of distributive posets....
This paper, based on recent research, articulates the opportunities and challenges posed by an emerging area of study known as ``mediumband wireless communication'', which refers to digital radio-frequency (RF) wireless communication through mediumba...
It is well known that the accuracy of a calibration depends strongly on the choice of camera poses from which images of a calibration object are acquired. We present a system -- Calibration Wizard -- that interactively guides a user towards taking op...
Estimating 3D hand and object pose from a single image is an extremely challenging problem: hands and objects are often self-occluded during interactions, and the 3D annotations are scarce as even humans cannot directly label the ground-truths from a...
With the increase number of companies focusing on commercializing Augmented Reality (AR), Virtual Reality (VR) and wearable devices, the need for a hand based input mechanism is becoming essential in order to make the experience natural, seamless and...
Web archives capture the history of the Web and are therefore an important source to study how societal developments have been reflected on the Web. However, the large size of Web archives and their temporal nature pose many challenges to researchers...
We pose the following conjecture: (*) If A is the union of line segments in R^n, and B is the union of the corresponding full lines then the Hausdorff dimensions of A and B agree. We prove that this conjecture would imply that every Besicovitch s...
This report is an introduction to mathematical map colouring and the problems posed by Heawood in his paper of 1890. There will be a brief discussion of the Map Colour Theorem; then we will move towards investigating empire maps in the plane and the...
In-memory computing architectures provide a much needed solution to energy-efficiency barriers posed by Von-Neumann computing due to the movement of data between the processor and the memory. Functions implemented in such in-memory architectures are...
We study the sharp interface limit and well-posedness of a phase field model for self-climb of prismatic dislocation loops in periodic settings. The model is set up in a Cahn-Hilliard/Allen-Cahn framework featured with degenerate phase-dependent diff...
Voice conversion (VC) using deep learning technologies can now generate high quality one-to-many voices and thus has been used in some practical application fields, such as entertainment and healthcare. However, voice conversion can pose potential so...
On the occasion of the 100th anniversary of the beginning of the revolutionary contributions to physics by Einstein, I am happy to respond to a problem posed by him in 1905. He said: In this paper it will be shown that according to the molecular-ki...
Processing long contexts poses a significant challenge for large language models (LLMs) due to their inherent context-window limitations and the computational burden of extensive key-value (KV) activations, which severely impact efficiency. For infor...
While artificial intelligence (AI) holds enormous promise, many experts in the field are warning that there is a non-trivial chance that the development of AI poses an existential threat to humanity. Existing regulatory initiative do not address this...
For a $(3+1)$-free poset $P$, we define a hybrid of $P$-tableaux and cylindric tableaux called cylindric $P$-tableaux. We introduce $P$-analogs of cylindric Schur functions, defined by a determinantal formula, and prove that they are the weight gener...
Recommender systems have received great commercial success. Recommendation has been used widely in areas such as e-commerce, online music FM, online news portal, etc. However, several problems related to input data structure pose serious challenge to...