B. Preliminaries B.1. Functions of Bounded Variation. We begin by discussing the concept of bounded variation. This concept will be used to formulate our assumptions on the distribution function F in the subsection immediately following this one, namely Section B.2. More importantly, it plays an essential role in our proof of sufficiency in a much later section, namely Section P. The simplest d...

Motivated by the path planning problem for robotic systems this paper considers nonholonomic path planning on the Euclidean group of motions SE(n) which describes a rigid bodies path in n-dimensional Euclidean space. The problem is formulated as a constrained optimal kinematic control problem where the cost function to be minimised is a quadratic function of translational and angular velocity i...

For the Schrödinger equation − d 2 u / x + q ( ) = λ on a finite x-interval, there is defined an “asymmetry function” ; , which entire of order 1/2 and type 1 in λ. Our main result identifies classes square-integrable potentials that possess common asymmetry function . any given one potential for each Dirichlet spectral sequence.

Analysis in general tends to revolve around the study of general classes of functions (often real-valued or complex-valued) and operators (which take one or more functions as input, and return some other function as output). Harmonic analysis focuses in particular on the quantitative properties of such functions, and how these quantitative properties change when apply various (often quite expli...

The bilinear maximal operator de ned below maps L L into L provided 1 < p; q <1, 1=p+ 1=q = 1=r and 2=3 < r 1. Mfg(x) = sup t>0 1 2t Z t t jf(x+ y)g(x y)j dy In particular Mfg is integrable(!) if f and, g are square integrable, answering a conjecture posed by Alberto Calder on. 1 Principal Results In 1964 Alberto Calder on de ned the maximal operator Mfg(x) = sup t>0 1 2t Z t t jf(x y)g(x y)j d...

We investigate the infinite-temperature dynamics of complex Sachdev-Ye-Kitaev model $({\mathrm{SYK}}_{4})$ complemented with a single-particle hopping term $({\mathrm{SYK}}_{2})$, leading to chaos-to-integrable crossover many-body eigenstates. Due presence all-to-all connected ${\mathrm{SYK}}_{2}$ term, nonequilibrium prethermal state emerges for finite time window ${t}_{\mathrm{th}}\ensuremath...

Arzelà's Bounded Convergence Theorem (1885) states that if a sequence of Riemann integrable functions on a compact interval is uniformly bounded and has an integrable pointwise limit function, then the sequence of their integrals tends to the integral of the limit. It is a trivial consequence of measure theory; in particular, of the Dominated Convergence Theorem. However, denying oneself the ma...

We study the spectral properties of a multiparametric system having particle-hole symmetry in random matrix setting. observe crossover from Poisson to Wigner-Dyson like behavior average local ratio spacing within spectrum single as function effective parameter referred complexity parameter. The varies logarithmically across transition. This is universal for different ensembles subjected same co...

1. Assume everything you know from the notes on Riemann-Stieltjes integration. (a) Show that if g : [0, 1]→ R is Riemann-integrable on [0, 1] and φ : [a, b]→ R is continuous on an interval [a, b] ⊇ g[[0, 1]], then φ ◦ g is Riemann-integrable on [0, 1]. (b) Assume that f : [0, 1]→ R is a function for which f ′(t) exists at each t ∈ [0, 1] (one-sidedly at endpoints) and that the Riemann integral ...