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Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture. (arXiv:2310.01450v1 [math.FA]) arxiv.org/abs/2310.01450

Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture

Let $(Ω, μ)$, $(Δ, ν)$ be measure spaces and $\{τ_α\}_{α\in Ω}$, $\{ω_β\}_{β\in Δ}$ be 1-bounded continuous Parseval frames for a Hilbert space $\mathcal{H}$. Then we show that \begin{align} (1) \quad \quad \quad \quad \log (μ(Ω)ν(Δ))\geq S_τ(h)+S_ω(h)\geq -2 \log \left(\frac{1+\displaystyle \sup_{α\in Ω, β\in Δ}|\langleτ_α, ω_β\rangle|}{2}\right) , \quad \forall h \in \mathcal{H}_τ\cap \mathcal{H}_ω, \end{align} where \begin{align*} &\mathcal{H}_τ:= \{h_1 \in \mathcal{H}: \langle h_1 , τ_α\rangle \neq 0, α\in Ω\}, \quad \mathcal{H}_ω:= \{h_2 \in \mathcal{H}: \langle h_2, ω_β\rangle \neq 0, β\in Δ\},\\ &S_τ(h):= -\displaystyle\int\limits_Ω\left|\left \langle \frac{h}{\|h\|}, τ_α\right\rangle \right|^2\log \left|\left \langle \frac{h}{\|h\|}, τ_α\right\rangle \right|^2\,dμ(α), \quad \forall h \in \mathcal{H}_τ, \\ & S_ω(h):= -\displaystyle\int\limits_Δ\left|\left \langle \frac{h}{\|h\|}, ω_β\right\rangle \right|^2\log \left|\left \langle \frac{h}{\|h\|}, ω_β\right\rangle \right|^2\,dν(β), \quad \forall h \in \mathcal{H}_ω. \end{align*} We call Inequality (1) as \textbf{Continuous Deutsch Uncertainty Principle}. Inequality (1) improves the uncertainty principle obtained by Deutsch \textit{[Phys. Rev. Lett., 1983]}. We formulate Kraus conjecture for 1-bounded continuous Parseval frames. We also derive continuous Deutsch uncertainty principles for Banach spaces.

arxiv.org

Enhancing Secrecy Capacity in PLS Communication with NORAN based on Pilot Information Codebooks. (arXiv:2310.01453v1 [eess.SP]) arxiv.org/abs/2310.01453

Enhancing Secrecy Capacity in PLS Communication with NORAN based on Pilot Information Codebooks

In recent research, non-orthogonal artificial noise (NORAN) has been proposed as an alternative to orthogonal artificial noise (AN). However, NORAN introduces additional noise into the channel, which reduces the capacity of the legitimate channel (LC). At the same time, selecting a NORAN design with ideal security performance from a large number of design options is also a challenging problem. To address these two issues, a novel NORAN based on a pilot information codebook is proposed in this letter. The codebook associates different suboptimal NORANs with pilot information as the key under different channel state information (CSI). The receiver interrogates the codebook using the pilot information to obtain the NORAN that the transmitter will transmit in the next moment, in order to eliminate the NORAN when receiving information. Therefore, NORAN based on pilot information codebooks can improve the secrecy capacity (SC) of the communication system by directly using suboptimal NORAN design schemes without increasing the noise in the LC. Numerical simulations and analyses show that the introduction of NORAN with a novel design using pilot information codebooks significantly enhances the security and improves the SC of the communication system.

arxiv.org

Wavelet-Harmonic Integration Methods. (arXiv:2310.01483v1 [hep-ph]) arxiv.org/abs/2310.01483

Wavelet-Harmonic Integration Methods

A new integration method drastically improves the efficiency of the dark matter direct detection calculation. In this work I introduce a complete, orthogonal basis of spherical wavelet-harmonic functions, designed for the new vector space integration method. This factorizes the numeric calculation into a part that depends only on the astrophysical velocity distribution; a second part, depending only on the detector form factor; and a scattering matrix defined on the basis functions, which depends on the details of the dark matter (DM) particle model (e.g. its mass). For common spin-independent DM-Standard Model interactions, this scattering matrix can be evaluated analytically in the wavelet-harmonic basis. This factorization is particularly helpful for the more complicated analyses that have become necessary in recent years, especially those involving anisotropic detector materials or more realistic models of the local DM velocity distribution. With the new method, analyses studying large numbers of detector orientations and DM particle models can be performed about 10 million times faster. This paper derives several analytic results for the spherical wavelets, including an extrapolation in the space of wavelet coefficients, and a generalization of the vector space method to a much broader class of linear functional integrals. Both results are highly relevant outside the field of DM direct detection.

arxiv.org

Artin approximation for left-right equivalence of map-germs and for quivers of map-germs. (arXiv:2310.01521v1 [math.AG]) arxiv.org/abs/2310.01521

Artin approximation for left-right equivalence of map-germs and for quivers of map-germs

The standard Artin approximation (AP) is not readily applicable to the left-right equivalence of map-germs Maps((k^n,o),(k^m,o)). Moreover, the naive extension does not hold in the k-analytic case, because of Osgood-Gabrielov-Shiota examples. The left-right version of Artin approximation (LRAP) was established by M. Shiota for map-germs that are either Nash or [real-analytic and of finite singularity type]. We establish LRAP for Maps(X,Y) where X,Y are k-analytic/k-Nash germs of schemes of any characteristic. More precisely: * (k-Nash power series, k<x>/J_X and k<y>/J_Y.) LRAP holds for any map. * (k-analytic power series, k{x}/J_X and k{y}/J_Y.) LRAP holds for maps of weakly-finite singularity type. This ``weakly-finite singularity type" (which we introduce) is a natural extension of the classical finite singularity type, and is of separate interest. As a trivial corollary we get: the inverse Artin approximation holds for analytic maps of weakly-finite singularity type. Then we extend the properties RAP, LAP, LRAP to the approximation results for quivers of maps, Gamma-AP. Finally, we establish the nested version, ``Gamma-AP with parameters". It is needed for families/unfoldings of maps.

arxiv.org

Derivation of a 2D PCCU-AENO method for nonconservative problems. Theory, Method and theoretical arguments. (arXiv:2310.00003v1 [math.NA]) arxiv.org/abs/2310.00003

Derivation of a 2D PCCU-AENO method for nonconservative problems. Theory, Method and theoretical arguments

In this paper, we introduce a methodology to design genuinely two-dimensional (2D) secondorder path-conservative central-upwind (PCCU) schemes. The scheme studies dam-break with high sediment concentration over abrupt moving topography quickly spatially variable even in the presence of resonance. This study is possible via a 2D sediment transport model (including arbitrarily sloping sediment beds and associated energy and entropy) in new generalized Shallow Water equations derived with associated energy and entropy in this work. We establish an existence theorem of global weak solutions. We show the convergence of a sequence of solutions of the proposed model. The second-order accuracy of the PCCU scheme is achieved using a new extension AENO (Averaging Essentially Non-Oscillatory) reconstruction developed in the 2D version of this work. We prove by rigorous demonstrations that the derived 2D scheme on structured meshes is well-balanced and positivity-preserving. Several tests are made to show the ability and superb performance of the proposed numerical modeling. The results obtained are compared with those existing in the literature and with experimental data. The current modeling improves some recent results in sediment transport and shows a good ability to simulate sediment transport in large-range environments.

arxiv.org

Semantic Communication with Probability Graph: A Joint Communication and Computation Design. (arXiv:2310.00015v1 [cs.IT]) arxiv.org/abs/2310.00015

Semantic Communication with Probability Graph: A Joint Communication and Computation Design

In this paper, we present a probability graph-based semantic information compression system for scenarios where the base station (BS) and the user share common background knowledge. We employ probability graphs to represent the shared knowledge between the communicating parties. During the transmission of specific text data, the BS first extracts semantic information from the text, which is represented by a knowledge graph. Subsequently, the BS omits certain relational information based on the shared probability graph to reduce the data size. Upon receiving the compressed semantic data, the user can automatically restore missing information using the shared probability graph and predefined rules. This approach brings additional computational resource consumption while effectively reducing communication resource consumption. Considering the limitations of wireless resources, we address the problem of joint communication and computation resource allocation design, aiming at minimizing the total communication and computation energy consumption of the network while adhering to latency, transmit power, and semantic constraints. Simulation results demonstrate the effectiveness of the proposed system.

arxiv.org

The extension problem for fractional powers of higher order of some evolutive operators. (arXiv:2310.00025v1 [math.AP]) arxiv.org/abs/2310.00025

The extension problem for fractional powers of higher order of some evolutive operators

This thesis studies the extension problem for higher-order fractional powers of the heat operator $H=Δ-\partial_t$ in $\mathbb{R}^{n+1}$. Specifically, given $s>0$ and indicating with $[s]$ its integral part, we study the following degenerate partial differential equation in the thick space $\mathbb{R}^{n+1}\times \mathbb{R}_y^+$, \begin{equation} \label{a:1} \mathscr{H}^{[s]+1}U= \left( \partial_{yy} +\frac{a}{y}\partial_y +H \right)^{[s]+1}U=0. \quad \quad (1) \end{equation} The connection between the Bessel parameter $a$ in (1) and the fractional parameter $s>0$ is given by the equation \begin{equation*} a= 1-2(s-[s]). \end{equation*} When $s\in(0,1)$ this equation reduces to the well-known relation $a=1-2s$, and in such case (1) becomes the famous Caffarelli-Silvestre extension problem. Generalising their result, in this thesis we show that the nonlocal operator $(-H)^{\,s}$ can be realised as the Dirichlet-to-Neumann map associated with the solution $U$ of the extension equation (1). In this thesis we systematically exploit the evolutive semigroup $\{P_τ^H \}_{τ>0}$, associated with the Cauchy problem \begin{equation*} \begin{cases} \partial_τu-Hu=0 u((x,t),0)=f(x,t). \end{cases} \end{equation*} This approach provides a powerful tool in analysis, and it has the twofold advantage of allowing an independent treatment of several complex calculations involving the Fourier transform, while at same time extending to frameworks where the Fourier transform is not available.

arxiv.org

Machine Learning Clifford invariants of ADE Coxeter elements. (arXiv:2310.00041v1 [cs.LG]) arxiv.org/abs/2310.00041

Machine Learning Clifford invariants of ADE Coxeter elements

There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for $A_8$, $D_8$ and $E_8$ for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output -- the invariants -- is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results.

arxiv.org

ShOpt.jl: A Julia Package for Empirical Point Spread Function Characterization of JWST NIRCam Data. (arXiv:2310.00071v1 [astro-ph.IM]) arxiv.org/abs/2310.00071

ShOpt.jl: A Julia Package for Empirical Point Spread Function Characterization of JWST NIRCam Data

As astronomical data grows in volume and complexity, the scalability of analysis software becomes increasingly important. At the same time, astrophysics analysis software relies heavily on open-source contributions, so languages and tools that prioritize both performance and readability are especially valuable. Julia, with its just-in-time compiler and high level syntax, offers a compelling alternative to traditional languages like Python or C. In this paper, we outline ShOpt.jl, a new software package for point spread function (PSF) characterization written in Julia. ShOpt.jl features a number of performance optimizations, such as multithreading, the use of preconditioners, and the implementation of the memory-limited Broyden-Fletcher-Goldfarb-Shanno algorithm, as well as the flexibility to choose between principal component analysis, an autoencoder, and analytic profiles for PSF characterization. As observatories like the James Webb Space Telescope bring astrophysics into a new era of wide-field, high-resolution imaging, the challenges of PSF modeling become more pronounced. Tools like ShOpt.jl provide the community with a scalable, efficient, and accurate solution to these challenges, while also demonstrating the potential of Julia as a language that meets the demands of modern astrophysical research.

arxiv.org
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