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Under suitable requirements on a kernel on a locally compact space, we develop a theory of inner (outer) pseudo-balayage of quite general signed Radon measures (not necessarily of finite energy) onto quite general sets (not necessarily closed). Such investigations were initiated in Fuglede's work (Anal. Math., 2016), which was, however, mainly concerned with the outer pseudo-balayage of positive measures of finite energy, whereas treatment of signed measures of infinite energy requires essentially new methods and approaches. To illustrate this, it is enough to note that in the particular case of kernels satisfying the balayage principle, the pseudo-balayage of positive measures coincides with their balayage, whereas for signed measures, these two concepts are drastically different. The theory thereby established enables us to provide necessary and sufficient conditions for the solvability of the inner Gauss variational problem, a point of interest for a number of recent researches in constructive function theory. This study covers many interesting kernels in classical and modern potential theory, which looks promising for further applications., Comment: 22 pages. arXiv admin note: text overlap with arXiv:2401.11043

Knowledge distillation is commonly employed to compress neural networks, reducing the inference costs and memory footprint. In the scenario of homogenous architecture, feature-based methods have been widely validated for their effectiveness. However, in scenarios where the teacher and student models are of heterogeneous architectures, the inherent differences in feature representation significantly degrade the performance of these methods. Recent studies have highlighted that low-frequency components constitute the majority of image features. Motivated by this, we propose a Low-Frequency Components-based Contrastive Knowledge Distillation (LFCC) framework that significantly enhances the performance of feature-based distillation between heterogeneous architectures. Specifically, we designe a set of multi-scale low-pass filters to extract the low-frequency components of intermediate features from both the teacher and student models, aligning them in a compact space to overcome architectural disparities. Moreover, leveraging the intrinsic pairing characteristic of the teacher-student framework, we design an innovative sample-level contrastive learning framework that adeptly restructures the constraints of within-sample feature similarity and between-sample feature divergence into a contrastive learning task. This strategy enables the student model to capitalize on intra-sample feature congruence while simultaneously enhancing the discrimination of features among disparate samples. Consequently, our LFCC framework accurately captures the commonalities in feature representation across heterogeneous architectures. Extensive evaluations and empirical analyses across three architectures (CNNs, Transformers, and MLPs) demonstrate that LFCC achieves superior performance on the challenging benchmarks of ImageNet-1K and CIFAR-100. All codes will be publicly available., Comment: 12 pages, 3 figures, conference paper

We show that the upper bounds for the $L^2$-norms of $L^1$-normalized quasimodes that we obtained in [9] are always sharp on any compact space form. This allows us to characterize compact manifolds of constant sectional curvature using the decay rates of lower bounds of $L^1$-norms of $L^2$-normalized log-quasimodes fully resolving a problem initiated by the second author and Zelditch [15]. We are also able to characterize such manifolds by the concentration of quasimodes near periodic geodesics as measured by $L^2$-norms over thin geodesic tubes., Comment: 16 pages

We show that the derived category of a locally compact Hausdorff space $X$ (subject to a mild technical condition) is smooth in the sense of non-commutative geometry if and only if $X$ is discrete and finite.

We describe the compact objects in the $\infty$-category of $\mathcal C$-valued sheaves $\text{Shv} (X,\mathcal C)$ on a hypercomplete locally compact Hausdorff space $X$, for $\mathcal C$ a compactly generated stable $\infty$-category. When $X$ is a non-compact connected manifold and $\mathcal C$ is the unbounded derived category of a ring, our result recovers a result of Neeman. Furthermore, for $X$ as above and $\mathcal C$ a nontrivial compactly generated stable $\infty$-category, we show that $\text{Shv} (X,\mathcal C)$ is compactly generated if and only if $X$ is totally disconnected., Comment: corrects Lemma 3.2

When a brane is moving in a compact space, bulk-probing signals originating at the brane can arrive back at the brane outside the lightcone of the emitting event. In this letter, we study how adiabatic perturbations in the brane fluid, coupled to a bulk fluid, propagate in the moving brane. In the non-dissipative regime, we find an effective sound speed for such perturbations, depending on the brane and bulk fluid energy densities, equations of state, and brane speed. In the tight-coupling approximation, the effective sound speed might be superluminal for brane and bulk fluids that satisfy the strong energy condition. This has immediate consequences for brane-world cosmology models., Comment: 15 pages, 3 figures

We study gauge theories and quantum gravity in a finite interval of time $\tau $, on a compact space manifold $\Omega $. The initial, final and boundary conditions are formulated in gauge invariant and general covariant ways by means of purely virtual extensions of the theories, which allow us to "trivialize" the local symmetries and switch to invariant fields (the invariant metric tensor, invariant quark and gluon fields, etc.). The evolution operator $U(t_{\text{f}},t_{\text{i}})$ is worked out diagrammatically for arbitrary initial and final states, as well as boundary conditions on $\partial \Omega $, and shown to be well defined and unitary for arbitrary $\tau =t_{\text{f}}-t_{\text{i}}<\infty $. We illustrate the basic properties in Yang-Mills theory on the cylinder., Comment: 52 pages; PRD

We provide a diagrammatic formulation of perturbative quantum field theory in a finite interval of time $\tau $, on a compact space manifold $\Omega $. We explain how to compute the evolution operator $U(t_{\text{f}},t_{\text{i}})$ between the initial time $t_{\text{i}}$ and the final time $t_{\text{f}}=t_{\text{i}}+\tau $, study unitarity and renormalizability, and show how to include purely virtual particles, by rendering some physical particles (and all the ghosts, if present) purely virtual. The details about the restriction to finite $\tau $ and compact $\Omega $ are moved away from the internal sectors of the diagrams (apart from the discretization of the three-momenta), and coded into external sources. So doing, the diagrams are as similar as possible to the usual $S$ matrix diagrams, and most known theorems extend straightforwardly. Unitarity is studied by means of the spectral optical identities, and the diagrammatic version of the identity $U^{\dag}(t_{\text{f}},t_{\text{i}})U(t_{\text{f}},t_{\text{i}})=1$. The dimensional regularization is extended to finite $\tau $ and compact $\Omega$, and used to prove, under general assumptions, that renormalizability holds whenever it holds at $\tau =\infty $, $\Omega =\mathbb{R}^{3}$. Purely virtual particles are introduced by removing the on-shell contributions of some physical particles, and the ghosts, from the core diagrams, and trivializing their initial and final conditions. The resulting evolution operator $U_{\text{ph}}(t_{\text{f}},t_{\text{i}})$ is unitary, but does not satisfy the more general identity $U_{\text{ph}}(t_{3},t_{2})U_{\text{ph}}(t_{2},t_{1})=U_{\text{ph}}(t_{3},t_{1})$. As a consequence, $U_{\text{ph}}(t_{\text{f}},t_{\text{i}})$ cannot be derived from a Hamiltonian in a standard way, in the presence of purely virtual particles., Comment: 66 pages; 7 figures; v2: JHEP

As miniaturized spacecraft (e.g., cubesats and smallsats) and instrumentation become an increasingly indispensable part of space exploration and scientific investigations, it is important to understand their potential susceptibility to space weather impacts resulting from the sometimes volatile space environment. There are multitude of complexities involved in how space environment interacts with different space hardware/electronics. Measurements of such impacts, however, have been lacking. Therefore, we recommend developing and/or procuring low-cost, low-power consumption, and compact sensor suites (mainly for space weather and impact purposes) and flying them on all future NASA (and U.S in general) missions in order to measure and quantify space weather impacts, in addition to the main instrumentation., Comment: White paper submitted to Decadal Survey for Solar and Space Physics (Heliophysics) 2024-2033 It can also be accessed here. http://surveygizmoresponseuploads.s3.amazonaws.com/fileuploads%2F623127%2F6920789%2F135-07722f5da5bdfe243452a894f743182b_ZhengYihua.pdf

We performed wave-optics-based numerical simulations at mid-infrared wavelengths to investigate how the presence or absence of entrance slits and optical aberrations affect the spectral resolving power $R$ of a compact, high-spectral-resolving-power spectrometer containing an immersion-echelle grating. We tested three cases of telescope aberration (aberration-free, astigmatism and spherical aberration), assuming the aberration budget of the Space Infrared Telescope for Cosmology and Astrophysics (SPICA), which has a 20-$\mathrm{\mu m}$-wavelength diffraction limit. In cases with a slit, we found that the value of $R$ at around 10--20 $\mathrm{\mu m}$ is approximately independent of the assumed aberrations, which is significantly different from the prediction of geometrical optics. Our results also indicate that diffraction from the slit improves $R$ by enlarging the effective illuminated area on the grating window and that this improvement decreases at short wavelengths. For the slit-less cases, we found that the impact of aberrations on $R$ can be roughly estimated using the Strehl ratio., Comment: Accepted for publication in the Journal of Astronomical Telescopes, Instruments, and Systems (JATIS)

Let $M=S^n/ \Gamma$ and $h$ be a nontrivial element of finite order $p$ in $\pi_1(M)$, where the integer $n, p\geq2$, $\Gamma$ is a finite abelian group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we prove that for every irreversible Finsler compact space form $(M,F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying \[ \frac{4p^2}{(p+1)^2} \big(\frac{\lambda}{\lambda+1} \big)^2 < K \leq 1,\;\;\lambda< \frac{p+1}{p-1}, \] there exist at least $n-1$ non-contractible closed geodesics of class $[h]$. In addition, if the metric $F$ is bumpy and \[ (\frac{4p}{2p+1})^2 (\frac{\lambda}{\lambda+1})^2 < K \leq 1,\;\;\lambda<\frac{2p+1}{2p-1}, \] then there exist at least $2[\frac{n+1}{2}]$ non-contractible closed geodesics of class $[h]$, which is the optimal lower bound due to Katok's example. For $C^4$-generic Finsler metrics, there are infinitely many non-contractible closed geodesics of class $[h]$ on $(M, F)$ if $\frac{\lambda^2}{(\lambda+1)^2} < K \leq 1$ with $n$ being odd, or $\frac{\lambda^2}{(\lambda+1)^2}\frac{4}{(n-1)^2} < K \leq 1$ with $n$ being even., Comment: 20 pages. arXiv admin note: text overlap with arXiv:1605.07292

In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface of bounded geometry in a generalized Robertson-Walker space-time. We prove that the flow preserves the space-likeness condition and exists for infinite time. We also prove convergence in the setting of manifolds with boundary. Our discussion generalizes previous work by Ecker, Huisken, Gerhardt and others with respect to a crucial aspects: we consider any non-compact Cauchy hypersurface under the assumption of bounded geometry. Moreover, we specialize the aforementioned works by considering globally hyperbolic Lorentzian space-times equipped with a specific class of warped product metrics., Comment: 51 pages

In this note we prove that a regular continuous open image of the Sorgenfrey line with an uncountable weight has a closed subspace that is homeomorphic to the Sorgenfrey line. As a corollary we deduce the theorem in the title.

We extend the definition of topological pressure to locally compact Hausdorff spaces, and we demonstrate a "variational principle" comparing the topological and measure theoretic pressures. Given a continuous $\mathbb{Z}_+^N$-action $T$ over a locally compact Hausdorff space $X$ and a continuous function vanishing at infinity $f \in C_0(X)$, we define topological pressure $P(T,f)$ using open covers of a special type we call "admissible covers". With this topological pressure, we demonstrate that \begin{equation*} P(T,f) = \sup_{\mu} P_\mu(T,f), \end{equation*} where the supremum is taken over all $T$-invariant probability Radon measures over $X$, and is equal to $0$ when there is none. In the last section, we present an example that illustrates why admissible covers are so adequate to deal with the non-compact case, while some other approaches would fail.

Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space. By analogy with the geometric models, and with the discrete coupon collector's problem and with cover times for finite Markov chains, one expects a "weak concentration" bound for the distribution of the cover time to hold under minimal assumptions. We give two such results, one for random fixed-radius balls and the other for sequentially arriving randomly-centered and deterministically growing balls. Each is in fact a simple application of a different more general bound, the former concerning coverage by i.i.d. random sets with arbitrary distribution, and the latter concerning hitting times for Markov chains with a strong monotonicity property. The growth model seems generally more tractable, and we record some basic results and open problems for that model., Comment: 16 pages

We develop a theory of inner balayage of a positive Radon measure $\mu$ of finite energy on a locally compact space $X$ to arbitrary $A\subset X$, generalizing Cartan's theory of Newtonian inner balayage on $\mathbb R^n$, $n\geqslant3$, to a suitable function kernel on $X$. As an application of the theory thereby established, we show that if the space $X$ is perfectly normal and of class $K_\sigma$, then a recent result by Bent Fuglede (Anal. Math., 2016) on outer balayage of $\mu$ to quasiclosed $A$ remains valid for arbitrary Borel $A$. We give in particular various alternative definitions of inner (outer) balayage, provide a formula for evaluation of its total mass, and prove convergence theorems for inner (outer) swept measures and their potentials. The results obtained do hold (and are new in part) for most classical kernels on $\mathbb R^n$, $n\geqslant2$, which is important in applications., Comment: 21 pages

We aim to investigate the infrared regime in the effective theory of the new quadratic gauge in the physical compact space by defining it on 4-sphere. Abelian dominance is characterized by off-diagonal gluons acquiring dynamical masses and it hints at existence of confinement. We do observe Abelian dominance in the ghost condensed vacuum of this theory on 4-sphere. Along with this observation, we find an unusual result that mass of an off-diagonal gluon on 4-sphere is position dependent in this theory as a consequence of the curved geometry. This suggests that the curvature of 4-sphere does not change the infrared behaviour of present theory on 4-sphere from that of the theory with the same quadratic gauge in Euclidean spacetime as Abelian dominance is observed in Euclidean spacetime too while it has caused mass of an off-diagonal gluon on 4-sphere to be position dependent. The effective action in the confined phase on 4-sphere and that in the 4-dim Euclidean space are found to be identical in a profound outcome., Comment: Published, 13 pages, 0 figures

We compute the discharging rate of a uniform electric field due to Schwinger pair production in $(1+1)$-dimensional scalar electrodynamics with a compact dimension of radius $R$. Our calculation is performed in real time, using the in-in formalism. For large compactification radii, $R\to\infty$, we recover the standard non compact space result. However, other ranges of values of $R$ and of the mass $m$ of the charged scalar give rise to a richer set of behaviors. For $R\gtrsim{\cal O}(1/m)$ with $m$ large enough, the electric field oscillates in time, whereas for $R\to 0$ it decreases in steps. We discuss the origin of these results., Comment: 13 pages, 5 figures

We consider a theory of scalar QED on a spatially compact 1+1-dimensional spacetime. By considering a constant electric field pointing down the compact dimension, we compute the quantum effective action by integrating out the scalar degrees of freedom in the Euclidean sector. Working in the saddle-point approximation we uncover two novel branches/physical regimes upon analytically continuing back to real time and discover a new result, hitherto unreported in previous literature. Implications of our results are discussed., Comment: Published version

Holographic duality implies that the geometric properties of the gravitational bulk theory should be encoded in the dual field theory. These naturally include the metric on dimensions that become compact near the conformal boundary, as is the case for any asymptotically locally $\text{AdS}_n\times\mathbb{S}^k$ spacetime. Almost all previous work on metric reconstruction ignores these dimensions and would thus at most apply to dimensionally-reduced metrics. In this work, we generalize the approach to bulk reconstruction using light-cone cuts and propose a prescription to obtain the full higher-dimensional metric of generic spacetimes up to an overall conformal factor. We first extend the definition of light-cone cuts to include information about the asymptotic compact dimensions, and show that the full conformal metric can be recovered from these extended cuts. We then give a prescription for obtaining these extended cuts from the dual field theory. The location of the usual cuts can still be obtained from bulk-point singularities of correlators, and the new information in the extended cut can be extracted by using appropriate combinations of operators dual to Kaluza-Klein modes of the higher-dimensional bulk fields., Comment: 32 pages, 6 figures; v2: minor clarifications and references added

Let $M=S^{2n+1}/ \Gamma$, $\Gamma$ is a finite group which acts freely and isometrically on the $(2n+1)$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we first investigate Katok's famous example about irreversible Finsler metrics on the spheres to study the topological structure of the contractible component of the free loop space on the compact space form $M$, then we apply the result to establish the resonance identity for homologically visible contractible minimal closed geodesics on every Finsler compact space form $(M, F)$ when there exist only finitely many distinct contractible minimal closed geodesics on $(M, F)$. As its applications, using this identity and the enhanced common index jump theorem for symplectic paths proved by Duan, Long and Wang in \cite{DLW2}, we show that there exist at least $2n+2$ distinct closed geodesics on every compact space form $S^{2n+1}/ \Gamma$ with a bumpy irreversible Finsler metric $F$ under some natural curvature condition, which is the optimal lower bound due to Katok's example., Comment: To appear in Calc. Var. and PDEs, arXiv admin note: text overlap with arXiv:1708.00857, arXiv:1607.02746

In this note, we investigate the electromagnetic radiation emitted from a revolving point charge in a compact space. If the point charge is circulating with an angular frequency $\omega_0$ on the $(x,y)$-plane at $z=0$ with boundary conditions, $x \sim x+2 \pi R$ and $y \sim y+2\pi R$, it emits radiation into the $z$-direction of $z$ in $ [-\infty, +\infty]$. We find that the radiation shows discontinuities as a function of $\omega_0 R$ at which a new propagating mode with a different Fourier component appears. For a small radius limit $\omega_0 R \ll 1$, all the Fourier modes except the zero mode on $(x,y)$-plane are killed, but an effect of squeezing the electric field totally enhances the radiation. In the large volume limit $\omega_0 R \rightarrow \infty$, the energy flux of the radiation reduces to the expected Larmor formula., Comment: to be published in PLA

Using Gottschalk's notion\,---\,weakly locally almost periodic point, we show in this paper that if $f\colon X\rightarrow X$ is a minimal continuous transformation of a compact Hausdorff space $X$ to itself, then for all entourage $\varepsilon$ of $X$, \begin{equation*} \inf_{x\in X}\left\{\liminf_{N-M\to\infty}\frac{1}{N-M}\sum_{n=M}^{N-1}1_{\varepsilon[x]}(f^nx)\right\}>0. \end{equation*} An analogous assertion also holds for each minimal $C^0$-semiflow $\pi\colon \mathbb{R}_+\times X\rightarrow X$ and for any minimal transformation group with discrete amenable phase group.

Let $M=S^n/ \Gamma$ and $h$ be a nontrivial element of finite order $p$ in $\pi_1(M)$, where the integer $n\geq2$, $\Gamma$ is a finite group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractible homologically visible minimal closed geodesics of the class $[h]$ on every Finsler compact space form $(M, F)$ when there exist only finitely many distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$ with a bumpy Finsler metric, which improves a result of Taimanov in [Taimanov 2016] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on $\mathcal{R}P^n$ in [arXiv:1607.02746] to general compact space forms., Comment: 33 pages, All comments are welcome. arXiv admin note: substantial text overlap with arXiv:1607.02746

Isotropic turbulence is typically studied numerically through the direct numerical simulations (DNS). The DNS flows are described by the Navier-Stokes equation in a 'box', defined through periodic boundary conditions. The DNS flows live in a compact space and they are not isotropic in their large scales. The investigation of important phenomena of isotropic turbulence, such as anomalous scaling, through the DNS is affected by large scale effects. In this work we put isotropic turbulence - or better, the associated formal theory - in a 'box', through imposing periodicity at the level of the correlations functions. We offer a framework where one may investigate isotropic theories/models through the data of DNS in a formally consistent manner. We work at the level of the Karman-Howarth equation. Unlike the Navier-Stokes equation, infinitely smooth periodicity is obstructed in this theory, a fact expressed by a sequence of relations obeyed by the normal modes of the Karman-Howarth equation. Similar relations are imparted to the two-point functions under the condition that the energy spectrum and energy transfer function are realizable. Naturally constructed closures scheme for the Karman-Howarth equation do not conform to such relations, thereby destroying realizability. A closure can be made to conform to a finite number of them by adding corrective terms, in a procedure which possesses certain analogies with the renormalization of quantum field theory. The spectrum becomes unphysical (through sign-changing oscillations) for infinitely large wavenumbers, but we can controllably extend the regime where the spectrum remains physical deep enough in the dissipation range so that to be realistically adequate. We show that one or two such 'regularity relations' are needed at most for comparisons of the predictions of the theory with the current resolution level results of the DNS., Comment: 57 pages, 3 figures

We study some topological spaces that can be considered as hyperspaces associated to noncommutative spaces. More precisely, for a NC compact space associated to a unital C*-algebra, we consider the set of closed projections of the second dual of the C*-algebra as the hyperspace of closed subsets of the NC space. We endow this hyperspace with an analog of Vietoris topology. In the case that the NC space has a quantum metric space structure in the sense of Rieffel we study the analogs of Hausdorff and infimum distances on the hyperspace. We also formulate some interesting problems about distances between sub-circles of a quantum torus., Comment: Keywords: C*-algebra, state space, closed projection, hyperspace, Vietoris topology, Hausdorff distance, infimum distance

Given a set of strings, the shortest common superstring problem is to find the shortest possible string that contains all the input strings. The problem is NP-hard, but a lot of work has gone into designing approximation algorithms for solving the problem. We present the first time and space efficient implementation of the classic greedy heuristic which merges strings in decreasing order of overlap length. Our implementation works in $O(n \log \sigma)$ time and bits of space, where $n$ is the total length of the input strings in characters, and $\sigma$ is the size of the alphabet. After index construction, a practical implementation of our algorithm uses roughly $5 n \log \sigma$ bits of space and reasonable time for a real dataset that consists of DNA fragments., Comment: 13 Pages, 3 figures, accepted to the 24th International Symposium on String Processing and Information Retrieval (SPIRE 2017)