arXiv subjects
Publications and source records attributed to Dae Gwan Lee.
Chebotarev's theorem on roots of unity states that all minors of a Fourier matrix are non-zero if and only if the order of the matrix is prime. We establish cases in which all principal minors of Fourier matrices of square-free order are non-zero. In a subsequent paper we discuss the case of composites containing squares.
For the $N$-dimensional Fourier matrix $\mathcal{F}_N$, we prove that if $N\geq 4$ is square-free, then every $2 \times 2$ and $3\times 3$ principal minor of $\mathcal{F}_N$ is nonzero. We also show that if $N\geq 4$ is not square-free, then $\mathcal{F}_N$ has zero principal minors of all sizes. Moreover, based on numerical experiments, we conjecture that if $N$ is square-free, then all principal minors of $\mathcal{F}_N$ are nonzero.
The proton-magnetic reaction is commonly used in MRI machines with a strong magnetic field of over 1 T, while this study hypothesized that the electron magnetic reaction of hydrogen could affect the hydrogen bonds of double-stranded DNA (dsDNA) at a low magnetic field below 0.01 T. The goal is to develop a hydrogen bonding magnetic reaction-based gene regulation (HBMR-GR) system. The polarities of DNA base pairs are derived from the relative electrostatic charge between purines and pyrimidines, which become positively and negatively charged, respectively. The Pyu dsDNAs with pyrimidine(s)-purine(s) sequences, ds3T3A, ds3C3G, and ds3C3A, showed stronger DNA hybridization potential, increased infrared absorption at 3400-3200 cm-1, and a unique DNA conformation in HPLC analysis compared to the corresponding Puy dsDNAs. To target the three-dimensional structure of dsDNA based on the DNA base pair polarities, one can use cyclic electromagnetic DNA simulation (CEDS) with approximately 25% efficiency for randomly oriented dsDNAs. CEDS was found to induce sequence-specific hybridization of target oligo-dsDNAs in 0.005M NaCl solution and sequence-specific conformation of oligo-dsDNAs in 0.1M NaCl solution. It was found that the Pyu oligo-dsDNAs were more responsible for the hybridization and conformational changes by CEDS than the Puy oligo-dsDNAs. CEDS decreased ethidium bromide (EtBr) DNA intercalation and spermidine DNA condensation depending on CEDS time in the binding assay. The results also included that the Pyu oligo-dsDNAs were more responsible for CEDS by forming stable and unique conformation of oligo-dsDNA than the Therefore, it is postulated that the low-level HBMR-based CEDS can enhance the hybridization potential of oligo-dsDNAs and subsequently lead to the unique DNA conformation required for the initiation of various DNA functions.
The existence of a Fourier basis with frequencies in $\mathbb{R}^d$ for the space of square integrable functions supported on a given parallelepiped in $\mathbb{R}^d$, has been well understood since the 1950s. In a companion paper, we derived necessary and sufficient conditions for a parallelepiped in $\mathbb{R}^d$ to permit an orthogonal basis of exponentials with frequencies constrained to be a subset of a prescribed lattice in $\mathbb{R}^d$, a restriction relevant in many applications. In this paper, we investigate analogous conditions for parallelepipeds that permit a Riesz basis of exponentials with the same constraints on the frequencies. We provide a sufficient condition on the parallelepiped for the Riesz basis case which directly extends one of the necessary and sufficient conditions obtained in the orthogonal basis case. We also provide a sufficient condition which constrains the spectral norm of the matrix generating the parallelepiped, instead of constraining the structure of the matrix.
We consider tilings $(\mathcal{Q},Φ)$ of $\mathbb{R}^d$ where $\mathcal{Q}$ is the $d$-dimensional unit cube and the set of translations $Φ$ is constrained to lie in a pre-determined lattice $A \mathbb{Z}^d$ in $\mathbb{R}^d$. We provide a full characterization of matrices $A$ for which such cube tilings exist when $Φ$ is a sublattice of $A\mathbb{Z}^d$ with any $d \in \mathbb{N}$ or a generic subset of $A\mathbb{Z}^d$ with $d\leq 7$. As a direct consequence of our results, we obtain a criterion for the existence of linearly constrained frequency sets, that is, $Φ\subseteq A\mathbb{Z}^d$, such that the respective set of complex exponential functions $\mathcal{E} (Φ)$ is an orthogonal Fourier basis for the space of square integrable functions supported on a parallelepiped $B\mathcal{Q}$, where $A, B \in \mathbb{R}^{d \times d}$ are nonsingular matrices given a priori. Similarly constructed Riesz bases are considered in a companion paper.
We develop new methods for constructing exponential Riesz bases by taking unions of exponential Riesz bases. These methods are based on taking unions of frequency sets and domains respectively and therefore allow easy construction. Together with examples that illustrate our methods, we also provide several examples showing the delicate nature of exponential Riesz bases.
We study the approximation properties of pseudo-differential operators with small time-frequency dispersion, meaning that their spreading functions are supported in a small neighborhood of the origin. It is commonly assumed that for such operators $H$, the output $Hf$ can differ only a little from a scalar multiple of the input $f$. However, we disprove this heuristic statement, hence revealing some unexpected behaviors of such operators.
The complex exponentials with integer frequencies form a basis for the space of square integrable functions on the unit interval. We analyze whether the basis property is maintained if the support of the complex exponentials is restricted to possibly overlapping subsets of the unit interval. We show, for example, that if $S_1, \ldots, S_K \subset [0,1]$ are finite unions of intervals with rational endpoints that cover the unit interval, then there exists a partition of $\mathbb{Z}$ into sets $Λ_1, \ldots, Λ_K$ such that $\bigcup_{k=1}^K \{ e^{2πi λ(\cdot)} χ_{S_k} : λ\in Λ_k \}$ is a Riesz basis for $L^2[0,1]$. Here, $χ_S$ denotes the characteristic function of $S$.
We consider Gabor Riesz sequences generated by a lattice $Λ\subset \mathbb{R}^2$ and a window function $g \in L^2(\mathbb{R})$ which is well localized in both time and frequency. When $g$ belongs to the Feichtinger algebra, we prove that only those time-frequency shifts with parameters from the lattice $Λ$ leave the corresponding Gabor space invariant. This improves on earlier results where only lattices of rational density were considered. A slightly weaker result is proved - again for lattices of general density - under the regularity assumptions of the classical Balian-Low theorem, where both $g$ and its Fourier transform belong to the Sobolev space $H^1(\mathbb{R})$. The proof relies on a combination of methods from time-frequency analysis and the theory of $C^\ast$-algebras, specifically the so-called irrational rotation algebra.
We prove that if $I_\ell = [a_\ell,b_\ell)$, $\ell=1, \ldots, L$, are disjoint intervals in $[0,1)$ with the property that the numbers $1, a_1, \ldots, a_L, b_1, \ldots, b_L$ are linearly independent over $\mathbb{Q}$, then there exist pairwise disjoint sets $Λ_\ell \subset \mathbb{Z}$, $\ell=1, \ldots, L$, such that for every $J \subset \{ 1, \ldots , L \}$, the system $\{e^{2πi λx} : λ\in \cup_{\ell \in J} \, Λ_\ell \}$ is a Riesz basis for $L^2 ( \cup_{\ell \in J} \, I_\ell)$. Also, we show that for any disjoint intervals $I_\ell$, $\ell=1, \ldots, L$, contained in $[1,N)$ with $N \in \mathbb{N}$, the orthonormal basis $\{e^{2πi n x} : n \in \mathbb{Z} \}$ of $L^2[0,1)$ can be complemented by a Riesz basis $\{e^{2πi λx} : λ\inΛ\}$ for $L^2(\cup_{\ell=1}^L \, I_{\ell})$ with some set $Λ\subset (\frac{1}{N} \mathbb{Z}) \backslash \mathbb{Z}$, in the sense that their union $\{e^{2πi λx} : λ\in \mathbb{Z} \cup Λ\}$ is a Riesz basis for $L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )$.
Despite the recent advances in the theory of exponential Riesz bases, it is yet unknown whether there exists a set $S \subset \mathbb{R}^d$ which does not admit a Riesz spectrum, meaning that for every $Λ\subset \mathbb{R}^d$ the set of exponentials $e^{2πi λ\cdot x}$ with $λ\inΛ$ is not a Riesz basis for $L^2(S)$. As a meaningful step towards finding such a set, we construct a set $S \subset [-\frac{1}{2}, \frac{1}{2}]$ which does not admit a Riesz spectrum containing a nonempty periodic set with period belonging in $α\mathbb{Q}_+$ for any fixed constant $α> 0$, where $\mathbb{Q}_+$ denotes the set of all positive rational numbers. In fact, we prove a slightly more general statement that the set $S$ does not admit a Riesz spectrum containing arbitrarily long arithmetic progressions with a fixed common difference belonging in $α\mathbb{N}$. Moreover, we show that given any countable family of separated sets $Λ_1, Λ_2, \ldots \subset \mathbb{R}$ with positive upper Beurling density, one can construct a set $S \subset [-\frac{1}{2}, \frac{1}{2}]$ which does not admit the sets $Λ_1, Λ_2, \ldots$ as Riesz spectrum. An interesting consequence of our results is the following statement. There is a set $V \subset [-\frac{1}{2}, \frac{1}{2}]$ with arbitrarily small Lebesgue measure such that for any $N \in \mathbb{N}$ and any proper subset $I$ of $\{ 0, \ldots, N-1 \}$, the set of exponentials $e^{2πi k x}$ with $k \in \cup_{n \in I} (N\mathbb{Z} {+} n)$ is not a frame for $L^2(V)$. The results are based on the proof technique of Olevskii and Ulanovskii in 2008.
We consider Gabor frames generated by a general lattice and a window function that belongs to one of the following spaces: the Sobolev space $V_1 = H^1(\mathbb R^d)$, the weighted $L^2$-space $V_2 = L_{1 + |x|}^2(\mathbb R^d)$, and the space $V_3 = \mathbb H^1(\mathbb R^d) = V_1 \cap V_2$ consisting of all functions with finite uncertainty product; all these spaces can be described as modulation spaces with respect to suitable weighted $L^2$ spaces. In all cases, we prove that the space of Bessel vectors in $V_j$ is mapped bijectively onto itself by the Gabor frame operator. As a consequence, if the window function belongs to one of the three spaces, then the canonical dual window also belongs to the same space. In fact, the result not only applies to frames, but also to frame sequences.
Let $\mathcal G\subset L^2(\mathbb R)$ be the subspace spanned by a Gabor Riesz sequence $(g,Λ)$ with $g\in L^2(\mathbb R)$ and a lattice $Λ\subset\mathbb R^2$ of rational density. It was shown recently that if $g$ is well-localized both in time and frequency, then $\mathcal G$ cannot contain any time-frequency shift $π(z) g$ of $g$ with $z\notinΛ$. In this paper, we improve the result to the quantitative statement that the $L^2$-distance of $π(z)g$ to the space $\mathcal G$ is equivalent to the Euclidean distance of $z$ to the lattice $Λ$, in the sense that the ratio between those two distances is uniformly bounded above and below by positive constants. On the way, we prove several results of independent interest, one of them being closely related to the so-called weak Balian-Low theorem for subspaces.
We present a Compressive Sensing algorithm for reconstructing binary signals from its linear measurements. The proposed algorithm minimizes a non-convex cost function expressed as a weighted sum of smoothed $\ell_0$ norms which takes into account the binariness of signals. We show that for binary signals the proposed algorithm outperforms other existing algorithms in recovery rate while requiring a short run time.
We study extra time-frequency shift invariance properties of Gabor spaces. For a Gabor space generated by an integer lattice, we state and prove several characterizations for its time-frequency shift invariance with respect to a finer integer lattice. The extreme cases of full translation invariance, full modulation invariance, and full time-frequency shift invariance are also considered. The results show a close analogy with the extra translation invariance of shift-invariant spaces.
The need of reconstructing discrete-valued sparse signals from few measurements, that is solving an undetermined system of linear equations, appears frequently in science and engineering. Whereas classical compressed sensing algorithms do not incorporate the additional knowledge of the discrete nature of the signal, classical lattice decoding approaches such as the sphere decoder do not utilize sparsity constraints. In this work, we present an approach that incorporates a discrete values prior into basis pursuit. In particular, we address unipolar binary and bipolar ternary sparse signals, i.e., sparse signals with entries in $\{0,1\}$, respectively in $\{-1,0,1\}$. We will show that phase transition takes place earlier than when using the classical basis pursuit approach and that, independently of the sparsity of the signal, at most $N/2$, respectively $3N/4$, measurements are necessary to recover a unipolar binary, and a bipolar ternary signal uniquely, where $N$ is the dimension of the ambient space. We will further discuss robustness of the algorithm and generalizations to signals with entries in larger alphabets.