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Author(s):  
Kamil Karaçuha ◽  
Vasil Tabatadze ◽  
Eldar Ismailovich Veliyev

An accurate hybrid method (numerical-analytical method) for the diffraction of H-polarized electromagnetic plane wave by perfectly electric conducting cylindrical bodies containing edges and a longitudinal slit aperture is proposed. This method is the combination of the Method of Moment and semi-inversion method. The current density function is expressed as the Chebyshev polynomials forming a complete orthogonal set of basis functions. Then, the initial problem is reduced to a system of linear algebraic equations. After inversion, the unknown coefficients are obtained. Then, near and far-field distributions, radar cross-sections are obtained. The resonances are observed for different values of the aperture size, radius of the arc, and the results are compared with previous outcomes.


2021 ◽  
Author(s):  
E. DeBenedictis ◽  
D. Söll ◽  
K. Esvelt

SummaryProtein translation using four-base codons occurs in both natural and synthetic systems. What constraints contributed to the universal adoption of a triplet-codon, rather than quadruplet-codon, genetic code? Here, we investigate the tolerance of the E. coli genetic code to tRNA mutations that increase codon size. We found that tRNAs from all twenty canonical isoacceptor classes can be converted to functional quadruplet tRNAs (qtRNAs), many of which selectively incorporate a single amino acid in response to a specified four-base codon. However, efficient quadruplet codon translation often requires multiple tRNA mutations, potentially constraining evolution. Moreover, while tRNAs were largely amenable to quadruplet conversion, only nine of the twenty aminoacyl tRNA synthetases tolerate quadruplet anticodons. These constitute a functional and mutually orthogonal set, but one that sharply limits the chemical alphabet available to a nascent all-quadruplet code. Our results illuminate factors that led to selection and maintenance of triplet codons in primordial Earth and provide a blueprint for synthetic biologists to deliberately engineer an all-quadruplet expanded genetic code.


Author(s):  
Yan V. Fyodorov ◽  
Wojciech Tarnowski

Abstract We study the distribution of the eigenvalue condition numbers $$\kappa _i=\sqrt{ ({\mathbf{l}}_i^* {\mathbf{l}}_i)({\mathbf{r}}_i^* {\mathbf{r}}_i)}$$ κ i = ( l i ∗ l i ) ( r i ∗ r i ) associated with real eigenvalues $$\lambda _i$$ λ i of partially asymmetric $$N\times N$$ N × N random matrices from the real Elliptic Gaussian ensemble. The large values of $$\kappa _i$$ κ i signal the non-orthogonality of the (bi-orthogonal) set of left $${\mathbf{l}}_i$$ l i and right $${\mathbf{r}}_i$$ r i eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite N expression for the joint density function (JDF) $${{\mathcal {P}}}_N(z,t)$$ P N ( z , t ) of $$t=\kappa _i^2-1$$ t = κ i 2 - 1 and $$\lambda _i$$ λ i taking value z, and investigate its several scaling regimes in the limit $$N\rightarrow \infty $$ N → ∞ . When the degree of asymmetry is fixed as $$N\rightarrow \infty $$ N → ∞ , the number of real eigenvalues is $$\mathcal {O}(\sqrt{N})$$ O ( N ) , and in the bulk of the real spectrum $$t_i=\mathcal {O}(N)$$ t i = O ( N ) , while on approaching the spectral edges the non-orthogonality is weaker: $$t_i=\mathcal {O}(\sqrt{N})$$ t i = O ( N ) . In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of N eigenvalues remain real as $$N\rightarrow \infty $$ N → ∞ . In such a regime eigenvectors are weakly non-orthogonal, $$t=\mathcal {O}(1)$$ t = O ( 1 ) , and we derive the associated JDF, finding that the characteristic tail $${{\mathcal {P}}}(z,t)\sim t^{-2}$$ P ( z , t ) ∼ t - 2 survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices.


Author(s):  
M. Akif Özkan ◽  
Burak Ok ◽  
Bo Qiao ◽  
Jürgen Teich ◽  
Frank Hannig

Abstract Writing programs for heterogeneous platforms optimized for high performance is hard since this requires the code to be tuned at a low level with architecture-specific optimizations that are most times based on fundamentally differing programming paradigms and languages. OpenVX promises to solve this issue for computer vision applications with a royalty-free industry standard that is based on a graph-execution model. Yet, the OpenVX ’ algorithm space is constrained to a small set of vision functions. This hinders accelerating computations that are not included in the standard. In this paper, we analyze OpenVX vision functions to find an orthogonal set of computational abstractions. Based on these abstractions, we couple an existing domain-specific language (DSL) back end to the OpenVX environment and provide language constructs to the programmer for the definition of user-defined nodes. In this way, we enable optimizations that are not possible to detect with OpenVX graph implementations using the standard computer vision functions. These optimizations can double the throughput on an Nvidia GTX GPU and decrease the resource usage of a Xilinx Zynq FPGA by 50% for our benchmarks. Finally, we show that our proposed compiler framework, called HipaccVX, can achieve better results than the state-of-the-art approaches Nvidia VisionWorks and Halide-HLS.


Fractals ◽  
2020 ◽  
Vol 28 (05) ◽  
pp. 2050091
Author(s):  
YANG-YANG XU ◽  
JING-CHENG LIU

Let the self-similar measure [Formula: see text] be generated by an expanding real matrix [Formula: see text] and a digit set [Formula: see text] in space [Formula: see text]. In this paper, we only consider [Formula: see text] and the case [Formula: see text] is similar. We show that there exists an infinite orthogonal set of exponential functions in [Formula: see text] if and only if [Formula: see text] for some [Formula: see text] with [Formula: see text]. Furthermore, for the cases that [Formula: see text] does not admit any infinite orthogonal set of exponential functions, the exact cardinality of orthogonal exponential functions in [Formula: see text] is given.


2020 ◽  
Vol 23 (3) ◽  
pp. 799-821
Author(s):  
Muhammad Ali ◽  
Sara Aziz ◽  
Salman A. Malik

AbstractInverse problem for a family of multi-term time fractional differential equation with non-local boundary conditions is studied. The spectral operator of the considered problem is non-self-adjoint and a bi-orthogonal set of functions is used to construct the solution. The operational calculus approach has been used to obtain the solution of the multi-term time fractional differential equations. Integral type over-determination condition is considered for unique solvability of the inverse problem. Different estimates of multinomial Mittag-Leffler functions alongside Banach fixed point theorem are used to prove the unique local existence of the solution of the inverse problem. Stability of the solution of the inverse problem on the given datum is established.


2020 ◽  
Vol 31 (08) ◽  
pp. 2050063
Author(s):  
Juan Su ◽  
Ming-Liang Chen

Let the self-affine measure [Formula: see text] be generated by an expanding matrix [Formula: see text] and a finite integer digit set [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] for an integer [Formula: see text], then [Formula: see text] admits an infinite orthogonal set of exponential functions if and only if there exists [Formula: see text] such that [Formula: see text] for some [Formula: see text] with [Formula: see text] and [Formula: see text].


2020 ◽  
Vol 32 (3) ◽  
pp. 673-681
Author(s):  
Ming-Liang Chen ◽  
Jing-Cheng Liu ◽  
Juan Su

AbstractLet the self-affine measure {\mu_{M,D}} be generated by an expanding real matrix {M=\operatorname{diag}(\rho_{1}^{-1},\rho_{2}^{-1})} and an integer digit set {D=\{(0,0)^{t},(\alpha_{1},\alpha_{2})^{t},(\beta_{1},\beta_{2})^{t}\}} with {\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\neq 0}. In this paper, the sufficient and necessary conditions for {L^{2}(\mu_{M,D})} to contain an infinite orthogonal set of exponential functions are given.


Fractals ◽  
2020 ◽  
Vol 28 (01) ◽  
pp. 2050016
Author(s):  
JUAN SU ◽  
ZHI-YONG WANG ◽  
MING-LIANG CHEN

For the self-affine measure [Formula: see text] generated by an expanding matrix [Formula: see text] and an integer digit set [Formula: see text] with [Formula: see text], Su et al. proved that if [Formula: see text], then [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if [Formula: see text] [J. Su, Y. Liu and J. C. Liu, Non-spectrality of the planar self-affine measures with four-element digit sets, Fractals (2019), https://doi.org/10.1142/S0218348X19501159 ]. In this paper, we show that the above conclusion also holds for [Formula: see text]. So, a complete characterization of [Formula: see text] containing an infinite orthogonal set of exponential functions is given.


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