Compact Case Research Articles

Rigidities of isoperimetric inequality under nonnegative Ricci curvature

The sharp isoperimetric inequality for non-compact Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions culminated by Balogh and Kristály (2023) also covering metric-measure spaces satisfying the nonnegative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterization of the equality cases is present in the literature. The scope of this paper is to settle this problem by proving, in the same generality as Balogh and Kristály (2023), that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone. Our result applies to different frameworks yielding as corollaries new rigidity results: it extends the rigidity results of Brendle (2023) for weighted Riemannian manifolds and the rigidity results of Antonelli et al. (2023) for general \mathsf{RCD} spaces. It also applies to the Euclidean setting by proving that optimizers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.

Max-Product Sampling Kantorovich Operators: Quantitative Estimates in Functional Spaces

In this paper, we study the order of approximation for max-product sampling Kantorovich operators based upon generalized kernels in the setting of Orlicz spaces. We establish a quantitative estimate for the considered family of sampling-type operators using the Orlicz-type modulus of smoothness, which involves the modular functional of the space. From this result, it is possible to obtain the qualitative order of convergence when functions belonging to suitable Lipschitz classes are considered. On the other hand, in the compact case, we exploit a suitable definition of K-functional in Orlicz spaces in order to provide an upper bound for the approximation error of the involved operators. The treatment in the general framework of Orlicz spaces allows one to obtain a unifying theory on the rate of convergence, as the proved results can be deduced for a wide range of functional spaces, such as L p -spaces, interpolation spaces and exponential spaces.

On invariant means and pre-syndetic subgroups

Beyond the locally compact case, equivalent notions of amenability diverge, and some properties no longer hold, for instance amenability is not inherited by topological subgroups. This investigation is guided by some amenability-type properties of groups of paths and loops. It is shown that a version of amenability called skew-amenability is inherited by pre-syndetic subgroups in the sense of Basso and Zucker (in particular, by co-compact subgroups). It follows that co-compact subgroups of amenable topological groups whose left and right uniformities coincide are amenable. We discuss a version of amenability belonging to P. Malliavin and M.-P. Malliavin: the existence of a mean on bounded Borel functions that is invariant under the left action of a dense subgroup. We observe that this property is in general strictly stronger than amenability, and establish for it Reiter- and Følner-type criteria. Finally, there is a review of open problems.

Sharp Sobolev inequalities on noncompact Riemannian manifolds with Ric≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{Ric}\\ge 0$$\\end{document} via optimal transport theory

In their seminal work, Cordero-Erausquin, Nazaret and Villani (Adv Math 182(2):307-332, 2004) proved sharp Sobolev inequalities in Euclidean spaces via Optimal Transport, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. By using L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-optimal transport approach, the compact case has been successfully treated by Cavalletti and Mondino (Geom Topol 21:603-645, 2017), even on metric measure spaces verifying the synthetic lower Ricci curvature bound. In the present paper we affirmatively answer the above question for noncompact Riemannian manifolds with non-negative Ricci curvature; namely, by using Optimal Transport theory with quadratic distance cost, sharp Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-Sobolev and Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-logarithmic Sobolev inequalities (both for p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document} and p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}) are established, where the sharp constants contain the asymptotic volume ratio arising from precise asymptotic properties of the Talentian and Gaussian bubbles, respectively. As a byproduct, we give an alternative, elementary proof to the main result of do Carmo and Xia (Math 140:818-826, 2004) and subsequent results, concerning the quantitative volume non-collapsing estimates on Riemannian manifolds with non-negative Ricci curvature that support Sobolev inequalities.

Emergence in string theory and Fermi gases

The Emergence Proposal suggests that some Swampland criteria, in particular on large field distances, are a consequence of the emergent nature of dynamics for fields in the infrared. In the context of type II string theory compactified on Calabi-Yau manifolds, it proposes that the cubic tree-level piece of the genus-zero prepotential is emergent from integrating out massive non-perturbative states. For a certain special non-compact Calabi-Yau, the blown-up conifold, it is known that the full all-genus prepotential can be matched onto the Grand Canonical potential of a two-dimensional Fermi gas. We propose here that this should be understood in the context of emergence: the prepotential is induced by integrating out the Fermi gas degrees of freedom. To make contact with the Swampland we need dynamical gravity, so compact Calabi-Yau manifolds. We show that for specifically the cubic term, an integrating out calculation also works for compact cases. In particular, the exact cubic term coefficient can be recovered from integrating out a Fermi gas for any compact Calabi-Yau that is an elliptic fibration over a reflexive toric base. We also propose a general map, for any one-parameter Calabi-Yau, between the Grand Canonical potential of the ultraviolet non-perturbative system and the period. In particular, this map leads to an emergent cubic term in the genus-zero prepotential for any such one-parameter model.

Further Development of a High Pulse-Energy LED at 4 kHz for Volumetric Particle Tracking Velocimetry and a Demonstration Using 32 Such LED Units

In 2022, we reported at the 20th Lisbon symposium on the development of 460 nm LED specifically designed for illumination in volumetric Particle Tracking Velocimetry (PTV). At that time, our pulsed LED unit emitted light with a 6 ◦ divergence beam and a top-hat power cross section, producing 20 µ s pulses at a 4 kHz repetition rate and a 0.15 mJ pulse energy. Since then, we have been improving the efficiency of this LED. Currently, our LED unit emits a beam with similar specifications but with a pulse energy up to 0.65 mJ , which is 4.3 times more than the previous LED unit. We improved the efficiency of the electrical driver by further decreasing the footprint of the SMD circuitry and by using higher quality electronic components. Additionally, we replaced the aluminium pin fin heat sink with a copper pin fin heat sink and we characterized the thermal dissipation. Using SolidWorks, we integrated the electric driver, the copper heat sink and the collimating optics into a compact mechanical casing. These improvements resulted in an LED unit with greater overall efficiency, a larger pulse energy and a volume footprint that is half the the size of the previous design. Furthermore, we constructed four batteries of these LEDs, with each battery containing eight LED units. Here, we report on our improvements to the LED unit and our considerations in its development. Additionally, we describe preliminary results of a 3D-PTV experiment with Helium Filled Soap Bubbles (HFSB), which were illuminated by the four LED batteries.

Sharp pinching theorems for complete submanifolds in the sphere

Abstract For every complete and minimally immersed submanifold f : M n → S n + p f\colon M^{n}\to\mathbb{S}^{n+p} whose second fundamental form satisfies | A | 2 ≤ n ⁢ p / ( 2 ⁢ p − 1 ) \lvert A\rvert^{2}\leq np/(2p-1) , we prove that it is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface in S 4 \mathbb{S}^{4} , thereby extending the well-known results by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete M n M^{n} . We also obtain the corresponding result for complete hypersurfaces with non-vanishing constant mean curvature, due to Alencar & do Carmo in the compact case, under the optimal bound on the umbilicity tensor. In dimension n ≤ 6 n\leq 6 , a pinching theorem for complete higher-codimensional submanifolds with non-vanishing parallel mean curvature is proved, partly generalizing previous work by Santos. Our approach is inspired by the conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni.

Immersions into Sasakian space forms

We study immersions of Sasakian manifolds into finite and infinite dimensional Sasakian space forms. After proving Calabi’s rigidity results in the Sasakian setting, we characterise all homogeneous Sasakian manifolds which admit a (local) Sasakian immersion into a nonelliptic Sasakian space form. Moreover, we give a characterisation of homogeneous Sasakian manifolds which can be embedded into the standard sphere both in the compact and noncompact case.

Social media, artificial intelligence (AI), and one burning question: How to balance problems and progress?

Research methodology This compact case study was developed from secondary sources readily available in the public domain. These secondary sources included websites, videos and articles. Case overview/synopsis Throughout 2023, social media companies faced a wide range of criticism on several fronts. Critics claimed that the companies were not doing enough to manage content and the algorithms were influencing American public opinion in the Israel-–Hamas war. Others argued that social media was negatively impacting the mental health of American youth. In response, the platforms reiterated their neutrality and emphasized the features, functions and policies that were designed to address the issues and encourage a positive user experience. As generative artificial intelligence (AI) grew in popularity, the impact on social media was inevitable. Was the convergence of social media and AI inspiring progress or exacerbating problems? How would society balance the opposing forces in a rapidly evolving environment? Complexity academic level This case should be used in marketing and management classes at the undergraduate level. Applicable concepts include AI, social media, content and information.

Characterization of invariant complex Finsler metrics on the complex Grassmann manifold

Let P:=U(p+q)/U(p)×U(q) be the complex Grassmann manifold and F:T1,0P→[0,+∞) be an arbitrary U(p+q)-invariant strongly pseudoconvex complex Finsler metric. We prove that F is necessary a Kähler-Berwald metric which is not necessary Hermitian quadratic. We also prove that F is Hermitian quadratic if and only if F is a constant multiple of the canonical U(p+q)-invariant Kähler metric on P. In particular on the complex projective space CPn=U(n+1)/U(n)×U(1), there exists no U(n+1)-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Fubini-Study metric. These invariant metrics are of particular interesting since they are the most important examples of strongly pseudoconvex complex Finsler metrics on P which are elliptic metrics in the sense that they enjoy very similar holomorphic sectional curvature and bisectional curvature properties as that of the U(p+q)-invariant Kähler metrics on P, nevertheless, these invariant metrics are not necessary Hermitian quadratic, hence provide nontrivial explicit examples for complex Finsler geometry in the compact cases.

Wigner- and Marchenko–Pastur-Type Limit Theorems for Jacobi Processes

AbstractWe study Jacobi processes $$(X_{t})_{t\ge 0}$$ ( X t ) t ≥ 0 on $$[-1,1]^N$$ [ - 1 , 1 ] N and $$[1,\infty [^N$$ [ 1 , ∞ [ N which are motivated by the Heckman–Opdam theory and associated integrable particle systems. These processes depend on three positive parameters and degenerate in the freezing limit to solutions of deterministic dynamical systems. In the compact case, these models tend for $$t\rightarrow \infty $$ t → ∞ to the distributions of the $$\beta $$ β -Jacobi ensembles and, in the freezing case, to vectors consisting of ordered zeros of one-dimensional Jacobi polynomials. We derive almost sure analogues of Wigner’s semicircle and Marchenko–Pastur limit laws for $$N\rightarrow \infty $$ N → ∞ for the empirical distributions of the N particles on some local scale. We there allow for arbitrary initial conditions, which enter the limiting distributions via free convolutions. These results generalize corresponding stationary limit results in the compact case for $$\beta $$ β -Jacobi ensembles and, in the deterministic case, for the empirical distributions of the ordered zeros of Jacobi polynomials. The results are also related to free limit theorems for multivariate Bessel processes, $$\beta $$ β -Hermite and $$\beta $$ β -Laguerre ensembles, and the asymptotic empirical distributions of the zeros of Hermite and Laguerre polynomials for $$N\rightarrow \infty $$ N → ∞ .

Quantification and Optimization of Compaction Energy Used in Earth Construction: Case of Static and Dynamic Compaction

Earth construction is a sustainable and environmentally friendly approach to building. In addition to their good thermal performance, earth materials are abundant, inexpensive, and readily available, reducing the need for resource-intensive materials like concrete and steel. Regarding the construction process of earth structures, which is based on compaction, there is often a difference between the laboratory compaction process and the onsite one. The energy consumed onsite to produce earth structures is still approximative and uncontrolled, which affects considerably the mechanical performances of earth walls. Then, the investigation of the optimal compaction energy is necessary. To optimize the on-site compaction energy used in rammed earth (RE), an experimental study is carried out to compare the dynamic compaction usually applied to produce RE walls to the static compaction using a mechanical press. By considering increasing dynamic and static energies, the physical and mechanical properties are analyzed for each case. The obtained results show that RE walls can be replaced by prefabricated pressed earth blocks where the compaction energy is reduced by 60% and the compressive strength is enhanced by 70% using static compaction, thus achieving 4 MPa without stabilization. This solution allows to reduce the execution time and to control the quality of earth buildings.

Environmental Quality bOX (EQ-OX): A Portable Device Embedding Low-Cost Sensors Tailored for Comprehensive Indoor Environmental Quality Monitoring.

The continuous monitoring of indoor environmental quality (IEQ) plays a crucial role in improving our understanding of the prominent parameters affecting building users' health and perception of their environment. In field studies, indoor environment monitoring often does not go beyond the assessment of air temperature, relative humidity, and CO2 concentration, lacking consideration of other important parameters due to budget constraints and the complexity of multi-dimensional signal analyses. In this paper, we introduce the Environmental Quality bOX (EQ-OX) system, which was designed for the simultaneous monitoring of quantities of some of the main IEQs with a low level of uncertainty and an affordable cost. Up to 15 parameters can be acquired at a time. The system embeds only low-cost sensors (LCSs) within a compact case, enabling vast-scale monitoring campaigns in residential and office buildings. The results of our laboratory and field tests show that most of the selected LCSs can match the accuracy required for indoor campaigns. A lightweight data processing algorithm has been used for the benchmark. Our intent is to estimate the correlation achievable between the detected quantities and reference measurements when a linear correction is applied. Such an approach allows for a preliminary assessment of which LCSs are the most suitable for a cost-effective IEQ monitoring system.

Non-perturbative topological string theory on compact Calabi-Yau 3-folds

We obtain analytic and numerical results for the non-perturbative amplitudes of topological string theory on arbitrary, compact Calabi-Yau manifolds. Our approach is based on the theory of resurgence and extends previous special results to the more general case. In particular, we obtain explicit trans-series solutions of the holomorphic anomaly equations. Our results predict the all orders, large genus asymptotics of the topological string free energies, which we test in detail against high genus perturbative series obtained recently in the compact case. We also provide additional evidence that the Stokes constants appearing in the resurgent structure are closely related to integer BPS invariants.

Stock buybacks in the spotlight

Theoretical Basis The theoretical basis for the case is information asymmetry and signaling theory, with buybacks providing a mechanism for reducing information asymmetry between management and investors. The controversy surrounding buybacks has led to political and regulatory scrutiny, which, consistent with evidence from academic research, may affect corporate behavior. Research methodology The compact case is based on secondary, public information about stock buybacks. All sources used are cited in-text, with full citations included in the references section at the end of the teaching note. Case Overview/Synopsis Stock buybacks, a means of providing returns to shareholders, have recently received increased scrutiny by politicians, media and shareholder activists. Proponents have argued that buybacks result in efficient allocation of capital by returning funds to shareholders, whereas opponents have criticized buybacks for enriching executives, providing tax advantages to shareholders and contributing to income inequality. Corporations did not curtail their use of buybacks after the Inflation Reduction Act of 2022 imposed an excise tax. The case frames the buyback debate in current events and focuses on the buyback activity of Apple. The case provides students the opportunity to analyze alternative ways that companies can provide returns to shareholders, evaluate impacts of buybacks on corporate stakeholders and appraise the reasons for, and implications of, current controversy regarding buybacks. Complexity/Academic Level This compact case is appropriate for upper-level undergraduate or graduate courses in financial accounting, tax and finance. This case provides an opportunity to analyze and evaluate stock buyback decisions in the context of the current controversy related to buybacks.

Axisymmetric Pressing Problem of Discrete Metal Materials

The paper presents an analytically closed solution to the problem of axisymmetric pressing of discrete metal materials by the method of jointly solving the differential equations of equilibrium of the metal and the plasticity conditions of the porous body, taking into account all pressing factors without exception: the type and properties of the charge, loading conditions, porosity, temperature, friction, etc. The purpose of this work is to develop the foundations of the engineering theory of pressure processing of discrete materials using the example of solving the problem of axisymmetric pressing of structurally inhomogeneous metal chips in a movable closed matrix. The basis for constructing a physical and mathema-tical model of the process is the idealized case of uniform compaction of a porous body with the subsequent determination of the lateral pressure coefficient corresponding to the actual degree of compaction at various stages of loading. The resulting equation for the relationship between the stress tensor components and the yield stress and relative compaction density represents the cylindrical Mises plasticity condition, which in the limit at zero porosity transforms into the plasticity condition for compact metals. The boundary value problem is solved for tangential stresses, taking into account the magnitude and direction of action of contact friction forces, which in their physical nature do not differ from the friction forces in the depth of the pressed material. The physico-mathematical model makes it possible to calculate the stress fields and density of the body according to the coordinates of the deformation zone, as well as energy-power parameters (pressure, force, work of deformation) provided that three structural and rheological characteristics are determined: the yield strength, relative compression and the degree of deformation compaction. Due to the fact that the problem is solved in relation to bodies of rotation in a general form and in a general formulation, the solution itself should be considered as methodological for any axisymmetric loading scheme.

Development of a cost-effective compact diode-laser-based photoacoustic sensing instrument for breast tissue diagnosis.

The photoacoustic (PA) technique, a noninvasive pump-probe technique, has found interesting applications in biomedical tissue diagnosis over the last decade. To take it a step further to clinical applications, the PA technique needs to be designed as an instrument focusing on a compact design, reducing the cost, and quickly providing a quantitative diagnosis. This work presents a design and characterization of a cost-effective, compact PA sensing instrument for biomedical tissue diagnosis. A compact laser diode case design is developed to house several laser diodes for PA excitation, and a pulsed current supply unit is also developed in-house to power the laser diodes to generate a 25ns current pulse at a frequency of 20kHz. After PA experimental data acquisition, the signal's frequency spectra were calculated to characterize the tissue quantitatively and correlated with their mechanobiological properties. The corresponding dominant frequency peak in the PA spectral response (PASR) study was low in the fibrofatty normal breast tissue , compared to the dominant frequency peak of in the fibrocystic disease tissue, which had increased glandular and stromal elements, thereby increased tissue density. The histopathological findings correlated with the PASR results, and the fibrocystic breast disease tissue exhibited a higher dominant frequency peak and energy compared to the normal breast tissue. We experimented with an in vitro PASR study of fibrocystic human breast tissues and successfully differentiated different tissue types using quantitative spectral parameters peak frequency, mean frequency, and spectral energy. This gives the potential to take this technique further for cost-effective and quick clinical applications.

Hindcast Insights from Storm Surge Forecasting of Super Typhoon Saola (2309) in Hong Kong with the Sea, Lake and Overland Surges from Hurricanes Model

Super Typhoon Saola (2309) skirted past south-southeast of Hong Kong within 40 km on the night of 1 September 2023, posing a significant storm surge threat to Hong Kong. Given the close proximity of Saola with a peak intensity of about 210 km/h within 300 km of Hong Kong, a close call of the “super typhoon direct-hit” scenario, this case provides valuable insights from a hindcast review of storm surge forecasts and warning operation using the Sea, Lake and Overland Surges from Hurricanes (SLOSH) model, which is the operational storm surge model adopted by the Hong Kong Observatory (HKO). The performance of the HKO’s PRobabilistic Inundation Map Evaluation System (PRIMES) using both statistical and model ensemble approaches was also reviewed in this paper. Saola was a challenging case for operational forecasting of a compact TC structure with changes in storm size and intensity when it came close to Hong Kong. With major observations of storm structure using weather radar and dense automatic weather station, tide gauge and water level gauge networks, the high sensitivity of storm surge forecasts to the storm size parameter and the distance of closest approach was clearly revealed in the case of Saola. Even with a circularly symmetric TC parametric model like SLOSH, the hindcast review results illustrated that the model outputs were reasonably accurate during the closest approach of Saola given an accurate storm size and distance of closest approach were input, and using a highly computationally efficient storm surge model made it possible for the nowcasting of storm surges to handle compact and intense TC direct-hit cases in operational TC forecasting. Taking a nowcasting approach not only helps provide more reliable storm tide forecasts, but also facilitates the formulation of a better warning strategy when making final-call decisions in emergency response actions, based on the more frequent real-time analysis of TC position, intensity and storm size and the more accurate prediction of these parameters. A nowcasting workflow for storm surge operation was proposed in this paper.

Hardware and software for automated examination of defects of hard tissues of teeth after endodontic intervention for fatigue and destruction

The work developed circuit engineering, design and software of the loading machine for automated examination of dental hard tissue defects after endodontics intervention for fatigue and destruction. The advantage of this development is the combination as cyclic loading methods simulating chewing movements and force effects on compression in a compact small-sized case, with low power consumption and small noise level. Thanks to the use of a screw transmission and a stepper motor in combination with a sensitive tensoresistive force sensor, it was possible to achieve high accuracy and resolution of 0.1 microns.
 A series of tests was conducted on real samples of endodontically treated teeth restored using fiberglass pins and cast metal stump inserts. It is shown that methods of restoring incisors and premolars of the upper jaw with the help of fiberglass pins have an advantage when the residual structure of the tooth is lacking due to a more uniform distribution of deformation stresses, since their elastic modulus is close to the elastic modulus of dentin.

On the space of null geodesics of a spacetime: the compact case, Engel geometry and retrievability

We compute the contact manifold of null geodesics of the family of spacetimes left{ left( mathbb {S}^2times mathbb {S}^1, g_circ -frac{d^2}{c^2}dt^2right) right} _{d,cin mathbb {N}^+text { coprime}}, with g_circ the round metric on mathbb {S}^2 and t the mathbb {S}^1-coordinate. We find that these are the lens spaces L(2c, 1) together with the pushforward of the canonical contact structure on STmathbb {S}^2cong L(2,1) under the natural projection L(2,1)rightarrow L(2c,1). We extend this computation to Ztimes mathbb {S}^1 for Z a Zoll manifold. On the other hand, motivated by these examples, we show how Engel geometry can be used to describe the manifold of null geodesics of a certain class of three-dimensional spacetimes, by considering the Cartan deprolongation of their Lorentz prolongation. We characterize the three-dimensional contact manifolds that are contactomorphic to the space of null geodesics of a spacetime. The characterization consists in the existence of an overlying Engel manifold with a certain foliation and, in this case, we also retrieve the spacetime.