Double Cover Research Articles

Perfect Pseudo-Matchings in Cubic Graphs

A perfect pseudo-matchingM in a cubic graph G is a spanning subgraph of G such that every component of M is isomorphic to K2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_2$$\\end{document} or to K1,3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{1,3}$$\\end{document}. In view of snarks G with dominating cycle C, this is a natural generalization of perfect matchings since G\\E(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G \\setminus E(C)$$\\end{document} is a perfect pseudo-matching. Of special interest are such M where G/M is planar because such G have a cycle double cover. We show that various well known classes of snarks contain planarizing perfect pseudo-matchings, and that there are at least as many snarks with planarizing perfect pseudo-matchings as there are cyclically 5-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$5-$$\\end{document}edge-connected snarks.

Examples of topologically unknotted tori

We show that certain smooth tori with group Z \mathbb {Z} in S 4 S^4 have exteriors with standard equivariant intersection forms, and so are topologically unknotted. These include the turned 1-twist-spun tori in the 4-sphere constructed by Boyle, the union of the genus one Seifert surface of Cochran and Davis that has no slice derivative with a ribbon disc, and tori with precisely four critical points whose middle level set is a 2-component link with vanishing Alexander polynomial. This gives evidence towards the conjecture that all Z \mathbb {Z} -surfaces in S 4 S^4 are topologically unknotted, which is open for genus one and two. It is unclear whether these tori are smoothly unknotted, except for tori with four critical points whose middle level set is a split link. The double cover of S 4 S^4 branched along any of these surfaces is a potentially exotic copy of S 2 × S 2 S^2 \times S^2 , and, in the case of turned twisted tori, we show they cannot be distinguished from S 2 × S 2 S^2 \times S^2 using Seiberg–Witten invariants.

Non-symplectic involutions of normal K3 surfaces associated with Hirzebruch surfaces

For a non-symplectic involution g of a K3 surface X, the fixed point set of g and the quotient space X/⟨g⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X/\\langle g\\rangle $$\\end{document} are well studied. In this paper, we study a non-symplectic involution f of a normal K3 surface Y such that the smooth model of Y is a double cover K3 surface of a Hirzebruch surface and f is induced by its covering involution. We determine the fixed point set of f, the singular points of Y, and the quotient space Y/⟨f⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y/\\langle f\\rangle $$\\end{document}.

K-stability of Casagrande–Druel varieties

Abstract We introduce a new subclass of Fano varieties (Casagrande–Druel varieties) that are 𝑛-dimensional varieties constructed from Fano double covers of dimension n − 1 n-1 . We conjecture that a Casagrande–Druel variety is K-polystable if the double cover and its base space are K-polystable. We prove this for smoothable Casagrande–Druel threefolds, and for Casagrande–Druel varieties constructed from double covers of P n − 1 \mathbb{P}^{n-1} ramified over smooth hypersurfaces of degree 2 ⁢ d 2d with n > d > n 2 > 1 n>d>\frac{n}{2}>1 . As an application, we describe the connected components of the K-moduli space parametrizing smoothable K-polystable Fano threefolds in the families № 3.9 and № 4.2 in the Mori–Mukai classification.

Counting circuit double covers

AbstractWe study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to for some ) instead of cycles (graphs with all degrees even). We give an almost‐exponential lower bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower bound for planar graphs. We conjecture that any bridgeless cubic graph has at least circuit double covers and we show an infinite class of graphs for which this bound is tight.

Investigating the performance of an affordable parabolic dish collector with double glass vacuum cavity cover

Parabolic dish collectors (PDC) are one of the most important ways to concentrate solar energy. In this study, the performance of a low-cost parabolic dish collector with high average output temperatures is investigated. The low-cost PDC has a 45 ° edge angle and a circular receiver. A vacuum double-glazed cavity is used to keep heat inside the receiver and reduce the rate of heat losses. On clear and cloudy days, tests were conducted with thermal oil as a heat transfer fluid (HTF) to analyze the performance of the PDC. The results show that the change in temperature of the heat transfer fluid between the open and closed ends of the cavity receivers affected the receiver heat gain, thermal efficiency, and exergy efficiency. It was also found that the HTF has the highest exit temperature of 221 °C, with an average radiation intensity of 1032 W/m2. With 48.4 % thermal efficiency and 9.1 % exergy efficiency, the results showed the best performance. When it came to total efficiency, PDC showed its highest value of 46.2 % at 13:17 p.m. on January 4th, 2024.

Five‐cycle double cover and shortest cycle cover

AbstractThe 5‐even subgraph cycle double cover conjecture (5‐CDC conjecture) asserts that every bridgeless graph has a 5‐even subgraph double cover. A shortest even subgraph cover of a graph is a family of even subgraphs which cover all the edges of and the sum of their lengths is minimum. It is conjectured that every bridgeless graph has an even subgraph cover with total length at most . In this paper, we study those two conjectures for weak oddness 2 cubic graphs and present a sufficient condition for such graphs to have a 5‐CDC containing a member with many vertices. As a corollary, we show that for every oddness 2 cubic graph satisfying the sufficient condition has a 4‐even subgraph ‐cover with total length at most . We also show that every oddness 2 cubic graph with girth at least 30 has a 5‐CDC containing a member of length at least and thus it has a 4‐even subgraph ‐cover with total length at most .

On the cubic Shimura lift to $\operatorname{PGL}(3)$: The fundamental lemma

The classical Shimura correspondence lifts automorphic representations on the double cover of \mathrm{SL}_{2} to automorphic representations on \mathrm{PGL}_{2} . Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of \mathrm{SL}_{3} to \mathrm{PGL}_{3} , and also characterize the image of the lift. The characterization would be through the non-vanishing of a certain global period involving a function in the space of the automorphic minimal representation \Theta_{\mathrm{SO}_8} for split \mathrm{SO}_{8}(\mathbb{A}) , consistent with a conjecture of Bump, Friedberg and Ginzburg (2001). In this paper, we first analyze a global distribution on \mathrm{PGL}_{3}(\mathbb{A}) involving this period and show that it is a sum of factorizable orbital integrals. The same is true for the Kuznetsov distribution attached to the triple cover of \mathrm{SL}_{3}(\mathbb{A}) . We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the fundamental lemma.

Visualization and Quantification of Facemask Leakage Flows and Interpersonal Transmission with Varying Face Coverings

Although mask-wearing is now widespread, the knowledge of how to quantify or improve their performance remains surprisingly limited and is largely based on empirical evidence. The objective of this study was to visualize the expiratory airflows from facemasks and evaluate aerosol transmission between two persons. Different visualization methods were explored, including the Schlieren optical system, laser/LED-particle imaging system, thermal camera, and vapor–SarGel system. The leakage flows and escaped aerosols were quantified using a hotwire anemometer and a particle counter, respectively. The results show that mask-wearing reduces the exhaled flow velocity from 2~4 m/s (with no facemask) to around 0.1 m/s, thus decreasing droplet transmission speeds. Cloth, surgical, and KN95 masks showed varying leakage flows at the nose top, sides, and chin. The leakage rate also differed between inhalation and exhalation. The neck gaiter has low filtration efficiency and high leakage fractions, providing low protection efficiency. There was considerable deposition in the mouth–nose area, as well as the neck, chin, and jaw, which heightened the risk of self-inoculation through spontaneous face-touching. A face shield plus surgical mask greatly reduced droplets on the head, neck, and face, indicating that double face coverings can be highly effective when a single mask is insufficient. The vapor–SarGel system provided a practical approach to study interpersonal transmission under varying close contact scenarios or with different face coverings.

2∞-Selmer rank parities via the Prym construction

We derive a local formula for the parity of the 2∞-Selmer rank of Jacobians of curves of genus 2 or 3 which admit an unramified double cover. We give an explicit example to show how this local formula gives rank parity predictions against which the 2-parity conjecture may be tested. Our results yield applications to the parity conjecture for semistable curves of genus 3.

The Prym Variety of a Dilated Double Cover of Metric Graphs

The Prym Variety of a Dilated Double Cover of Metric Graphs

The centre of the Dunkl total angular momentum algebra

For a finite dimensional representation V of a finite reflection group W, we consider the rational Cherednik algebra Ht,c(V,W) associated with (V,W) at the parameters t≠0 and c. The Dunkl total angular momentum algebra Ot,c(V,W) arises as the centraliser algebra of the Lie superalgebra osp(1|2) containing a Dunkl deformation of the Dirac operator, inside the tensor product of Ht,c(V,W) and the Clifford algebra generated by V.We show that when dim⁡V≥3 and for every value of the parameter c, the centre of Ot,c(V,W) is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not (−1)V is an element of the group W. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into Ot,c(V,W), we establish results analogous to “Vogan's conjecture” for a family of operators depending on suitable elements of the double cover W˜.

Thermal assessment of cylindrical parabolic integrated collector storage using input-output and dynamic system testing procedures: Experimental and numerical study

The main limitation of Integrated Collector Storage systems lies in their low efficiencies and high loss coefficients. In this paper, experimental and numerical setups are conducted to assess the thermal performances of low cost Cylindrical Parabolic Integrated Collector storage (CP-ICS). The conceived system has two aluminum plates in parabolic form serving as reflectors, each with a surface area of 2 m2. The storage tank has a volume of 160 L covered with a layer of black paint with single and double transparent insulations. Results of the experimental tests using Input-Output method showed that the daily thermal efficiency ηd of the developed systems is equal to 48.21% and 49.46% for single and double insulation cover configurations, respectively. The total store heat capacity Cs, the useful collector surface Ac* and the storage tank heat losses coefficient Us of the system found using Dynamic System Testing procedure are equal to 0.56 MJ/K, 0.74 m2 and 1.59 W/K, respectively. Even if the energy efficiency of the system is slightly lower than that recorded in conventional systems, numerical results of long-term study using TRNSYS software showed that the system provides a reasonable solar fraction for the needs of a family in Tunisian climate. A comparative assessment of the developed solar collector performances in different representative climates showed that the use of the CP-ICS system presents a promising solution for countries with annual ambient temperatures fluctuating from 13°C to 33°C, such as Araxos with a solar coverage of 30.65% for a daily supply volume of 160 L. More importantly, in Faya-Largeau location, presenting Chadian climate data, the solar fraction is found to be the highest and reached an average of 67.25%.

Mirror symmetry for double cover Calabi–Yau varieties

Mirror symmetry for double cover Calabi–Yau varieties

Towards geodesic ridge curve for region-wise linear representation of geodesic distance field

This paper addresses the challenge of representing geodesic distance fields on triangular meshes in a piecewise linear manner. Unlike general scalar fields, which often assume piecewise linear changes within each triangle, geodesic distance fields pose a unique difficulty due to their non-differentiability at ridge points, where multiple shortest paths may exist. An interesting observation is that the geodesic distance field exhibits an approximately linear change if each triangle is further decomposed into sub-regions by the ridge curve. However, computing the geodesic ridge curve is notoriously difficult. Even when using exact algorithms to infer the ridge curve, desirable results may not be achieved, akin to the well-known medial-axis problem. In this paper, we propose a two-stage algorithm. In the first stage, we employ Dijkstra's algorithm to cut the surface open along the dual structure of the shortest path tree. This operation allows us to extend the surface outward (resembling a double cover but with distinctions), enabling the discovery of longer geodesic paths in the extended surface. In the second stage, any mature geodesic solver, whether exact or approximate, can be employed to predict the real ridge curve. Assuming the fast marching method is used as the solver, despite its limitation of having a single marching direction in a triangle, our extended surface contains multiple copies of each triangle, allowing various geodesic paths to enter the triangle and facilitating ridge curve computation. We further introduce a simple yet effective filtering mechanism to rigorously ensure the connectivity of the output ridge curve. Due to its merits, including robustness and compatibility with any geodesic solver, our algorithm holds great potential for a wide range of applications. We demonstrate its utility in accurate geodesic distance querying and high-fidelity visualization of geodesic iso-lines.

Nodal quintic surfaces and lines on cubic fourfolds (with an appendix by John Christian Ottem)

We study nodal quintic surfaces with an even set of 16 nodes as analogues of singular Kummer surfaces. The interpretation of the natural double cover of an even 16 -nodal quintic as a certain Fano variety of lines could be viewed as a replacement for the additive structure of the cover of a singular Kummer surface by its associated abelian surface. Most of the results in this article can be seen as refinements of known facts and our arguments rely heavily on techniques developed by Beauville (1979), Murre (1972), and Voisin (1986), Results due to Shen (2012, 2014) are particularly close to some of the statements. In this sense, the text is mostly expository (but with complete proofs), although our arguments often differ substantially from the original sources.

Matrices for finite group representations that respect Galois automorphisms

AbstractWe are given a finite group H, an automorphism $$\tau $$ τ of H of order r, a Galois extension L/K of fields of characteristic zero with cyclic Galois group $$\langle \sigma \rangle $$ ⟨ σ ⟩ of order r, and an absolutely irreducible representation $$\rho :H\rightarrow \textsf {GL} (n,L)$$ ρ : H → GL ( n , L ) such that the action of $$\tau $$ τ on the character of $$\rho $$ ρ is the same as the action of $$\sigma $$ σ . Then the following are equivalent. $$\bullet $$ ∙ $$\rho $$ ρ is equivalent to a representation $$\rho ':H\rightarrow \textsf {GL} (n,L)$$ ρ ′ : H → GL ( n , L ) such that the action of $$\sigma $$ σ on the entries of the matrices corresponds to the action of $$\tau $$ τ on H, and $$\bullet $$ ∙ the induced representation $$\textsf {ind} _{H,H\rtimes \langle \tau \rangle }(\rho )$$ ind H , H ⋊ ⟨ τ ⟩ ( ρ ) has Schur index one; that is, it is similar to a representation over K. As examples, we discuss a three dimensional irreducible representation of $$A_5$$ A 5 over $$\mathbb {Q}[\sqrt{5}]$$ Q [ 5 ] and a four dimensional irreducible representation of the double cover of $$A_7$$ A 7 over $$\mathbb {Q}[\sqrt{-7}]$$ Q [ - 7 ] .

Topological Strings on Non-commutative Resolutions

In this paper we propose a definition of torsion refined Gopakumar–Vafa (GV) invariants for Calabi–Yau threefolds with terminal nodal singularities that do not admit Kähler crepant resolutions. Physically, the refinement takes into account the charge of five-dimensional BPS states under a discrete gauge symmetry in M-theory. We propose a mathematical definition of the invariants in terms of the geometry of all non-Kähler crepant resolutions taken together. The invariants are encoded in the A-model topological string partition functions associated to non-commutative (nc) resolutions of the Calabi–Yau. Our main example will be a singular degeneration of the generic Calabi–Yau double cover of P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^3$$\\end{document} and leads to an enumerative interpretation of the topological string partition function of a hybrid Landau–Ginzburg model. Our results generalize a recent physical proposal made in the context of torus fibered Calabi–Yau manifolds by one of the authors and clarify the associated enumerative geometry.

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

Let N be a connected nonorientable surface with or without boundary and punctures, and j\colon S\rightarrow N be the orientation double covering. It has previously been proved that j induces an embedding \iota\colon\mathrm{Mod}(N)\hookrightarrow\mathrm{Mod}(S) with one exception. In this paper, we prove that the injective homomorphism \iota is a quasi-isometric embedding. The proof is based on the semihyperbolicity of \mathrm{Mod}(S) , which has already been established. We also prove that the embedding \mathrm{Mod}(F') \hookrightarrow \mathrm{Mod}(F) induced by an inclusion of a pair of possibly nonorientable surfaces F' \subset F is a quasi-isometric embedding.

An alliance of evidence-based decision-makers against populism

In a double cover interview at the start of 2024 with ‚Monitor Health Services Research‘, Prof Josef Hecken, the impartial chairman of the Federal Joint Committee (G-BA), and Prof Wolfgang Hoffmann, Chairman of the German Network for Health Services Research (DNVF), discuss the future role, significance and position of health services research, which has now come of age. The more frequently its results are incorporated into far-reaching political decision-making processes - such as the current federal hospital reform - the more headwinds are blowing against it. Health services research must learn to deal with this.