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Research Paper

Applied Science and Convergence Technology 2018; 27(5): 90-94

Published online September 30, 2018

https://doi.org/10.5757/ASCT.2018.27.5.90

Gapped Nearly Free-Standing Graphene on an SiC(0001) Substrate Induced by Manganese Atoms

Jinwoong Hwanga, Ji-Eun Leea, Minhee Kanga, Byeong-Gyu Parkb, Jonathan Denlingerc, Sung-Kwan Moc, and Choongyu Hwanga,*

aDepartment of Physics, Pusan National University, Busan 46241, Korea, bPohang Accelerator Laboratory, Pohang University of Science and Technology, Pohang 37673, Korea, cAdvanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

Correspondence to:*E-mail: ckhwang@pusan.ac.kr

Received: September 17, 2018; Revised: September 28, 2018; Accepted: September 29, 2018

The electron band structure of manganese-adsorbed graphene on an SiC(0001) substrate has been studied using angle-resolved photoemission spectroscopy. Upon introducing manganese atoms, the conduction band of graphene, that is observed in pristine graphene indicating intrinsic electron-doping by the substrate, completely disappears and the valence band maximum is observed at 0.4 eV below Fermi energy. At the same time, the slope of the valence band decreases by the presence of manganese atoms, approaching the electron band structure calculated using the local density approximation method. The former provides experimental evidence of the formation of nearly free-standing graphene on an SiC substrate, concomitant with a metal-to-insulator transition. The latter suggests that its electronic correlations are efficiently screened, suggesting that the dielectric property of the substrate is modified by manganese atoms and indicating that electronic correlations in grpahene can also be tuned by foreign atoms. These results pave the way for promising device application using graphene that is semiconducting and charge neutral.

Keywords: Graphene, SiC, Manganese, Free-standing, Metal-to-insulator transition, Angle-resolved photoemission spectroscopy

Graphene, consisting of a single atomic layer of carbon atoms arranged in a honeycomb lattice, has attracted a lot of research interest due to its two-dimensional nature of Dirac particles realized in a condensed matter system [1,2]. Moreover, graphene is expected to have great potential for fabrication of electronic devices with unique functionalities [3,4]. However, in practice, its gapless semi-metallic nature [5] and inevitable substrate effects, such as hybridization and charge transfer [69], make it difficult to utilize graphene as a building block for device applications, while preserving the characteristic properties of free-standing graphene. Therefore, decoupling of graphene from the substrate, especially aligning Dirac energy, ED, with Fermi energy, EF, and opening of an energy gap at ED have been two of the key objectives in graphene research [921].

The pursuit for decoupled and gapped graphene has led to exploring the introduction of foreign atoms. Intercalation of foreign atoms breaks the bonding between graphene and the substrate, or compensates for the work function difference between them, leading to transfer of electrons to the substrate that shifts ED towards EF [915]. The presence of a substrate can lift the sublattice symmetry of graphene, opening an energy gap at ED [1618]. Intercalation of foreign atoms further breaks sublattice symmetry, leading to enlargement of the energy gap [1921]. However, the alignment of ED with EF in conjunction with the opening of an energy gap at ED has been barely rarely reported so far.

Graphene epitaxially grown on an SiC substrate is a promising candidate for this purpose. The energy gap at ED is manipulated by the amount of foreign atoms [1921]. While ED lies far below EF, graphene gains in metallic properties. On the other hand, the interfacial layer of the SiC substrate is significantly modified by chemical bonding between foreign atoms and silicon atoms in the substrate, such that electrons are transferred back to the substrate, resulting in ED alignment with EF [1315].

In this study, angle-resolved photoemission spectroscopy (ARPES) has been utilized to investigate graphene on an SiC substrate. The presence of manganese atoms in the system not only shifts ED to EF, but also leads to a metalto-insulator transition, resulting in gapped nearly free-standing graphene on the SiC substrate. In addition, the observed graphene π band is agreement with the calculated graphene π band for the case when electron-electron interaction is efficiently screened, indicating the possibility that manganese atoms form a silicide layer with silicon atoms of the SiC substrate, making the interfacial layer of the substrate metallic. These findings suggest that the introduction of transition metals in graphene grown on an SiC substrate provides a viable approach towards the realization of graphene-based electronic devices preserving the characteristic properties of free-standing graphene.

II. Experimental details

Graphene samples were epitaxially grown on a 6HSiC(0001) substrate under silicon flux in a high vacuum chamber with a base pressure of 3 × 10−8 Torr. Manganese atoms were thermally evaporated by resistively heating a tungsten coil wrapped around a small piece of high-purity manganese (99.9 %), followed by another resistive heating of the graphene sample at 600 °C in order to promote intercalations of the deposited metal atoms [22]. Prior to ARPES measurements, the sample was heated to 700 °C to remove air contaminants in a sample preparation chamber with a base pressure of 1 × 10−10 Torr connected to the ARPES chamber. ARPES measurements were performed at Beamline 4A1 of the Pohang Acceleration Laboratory, and Beamlines 4.0.3 and 10.0.1 of the Advanced Light Source, Lawrence Berkeley National Laboratory. ARPES data were taken at 100 K with the energy and angular resolutions of 40 meV and 0.1°, respectively.

Figures 1(a) and 1(b) show ARPES intensity maps of as-grown graphene taken parallel and perpendicular to the ΓK direction of the graphene unit cell, respectively, as denoted by red lines in the insets. The ARPES intensity maps show the characteristic conical energy-momentum dispersion of graphene with two almost parallel dispersions of strong and weak spectral intensity denoted by blue and red arrows, respectively. The pseudospin of quasiparticles in graphene results in a photoemission matrix element effect in which photoelectron intensity is strongly suppressed along the ΓK direction [23], such that one of the branches of the conical dispersion is not observed as shown in Fig. 1(a). The strong and weak spectral intensities correspond to single- and double-layer graphene, respectively. Since photoelectron intensity is closely related to the scattering area of a sample, the strong spectral intensity indicates that single-layer graphene is dominant in the sample. These spectral features are also illustrated in the energy distribution curves (EDCs) shown in Figs. 1(e) and 1(f).

In the measured ARPES intensity map for as-grown graphene, ED, denoted by a black arrow is observed at 0.4 eV below EF, indicating intrinsic electron doping in the presence of the SiC substrate [24]. One can notice that the upper conical dispersion, the so-called π* band or conduction band, and the lower conical dispersion, the so-called π band or valence band, do not merge at a single point. The merging point is elongated along the energy direction, leaving a high spectral intensity around ED [25], inconsistent with the zero density of states of ideal graphene [5]. A carbidic layer, the so-called buffer layer, exists in between graphene and the SiC substrate [26], and exhibits the same geometric structure as graphene in the absence of the conical dispersion due to the formation of covalent bonds with the substrate. The presence of the buffer layer breaks sublattice symmetry in single-layer graphene, leading to the opening of an energy gap at ED [16,18]. The buffer layer also leads to an in-gap state, that is observed only for single-layer graphene on an SiC substrate [18,25]. These two features are responsible for the elongated spectrum at ED observed in Fig. 1(b).

The introduction of manganese atoms into the system makes three notable changes in the electron band structure of graphene as shown in Figs. 1(c) and 1(d). A non-dispersive band emerges at 2.8 eV below EF as denoted by the black dashed line. It is not observed in as-grown graphene shown in Figs. 1(a) and 1(b), but bears similarity to the localized d band observed in graphene with transition metals such as nickel, cobalt, and iron [7,9,11]. The non-dispersive band is well resolved in the EDCs shown in Figs. 1(g) and 1(h), confirming the presence of manganese atoms in the system. Moreover, it is also important to note that the conduction band observed in as-grown graphene completely disappears, indicating that charge neutrality of free-standing graphene is recovered although graphene is standing on the SiC substrate. Finally, the valence band maximum is observed at ~0.4 eV below EF as shown in Fig. 1(d) and denoted by the blue dashed-line in Fig. 1(h), while the spectral intensity between the valence band maximum and EF is strongly suppressed. When the difference between the valence band maximum and EF is typically half the energy gap, the energy gap of graphene on an SiC substrate with manganese atoms can be as much as ~0.8 eV. In other words, the charge carrier density of ~1013 cm−2 of as-grown graphene is significantly reduced, so that metallic graphene becomes a band insulator. In epitaxial graphene on an SiC substrate, the presence of the substrate breaks the sublattice symmetry of graphene, such that the conduction and valence bands of graphene exhibit a finite separation of ≤0.26 eV [18]. The larger energy gap observed in this system indicates that the sublattice symmetry of graphene might be further lifted by the presence of manganese atoms.

The electronic structures observed in graphene on an SiC substrate without and with manganese atoms and their geometric structures are summarized in Fig. 2. The metal-to-insulator transition concomitant with the shift of ED to EF bears two possibilities. First, manganese atoms can induce hole doping in graphene, compensating electron doping by the substrate. However, a previous scanning tunneling spectroscopy study shows that manganese atoms induce electron doping [22], excluding this possibility. Epitaxial graphene on an SiC(0001) substrate is electron-doped due to the formation of a Schottky barrier in between graphene and the substrate [24]. The Schottky barrier can be lowered, transferring charge back to the substrate, when manganese atoms modify the geometric structure of the interfacial layer of the substrate. Indeed, when gold [13] or tin [15] atoms are introduced in graphene on an SiC substrate, these atoms chemically bond with silicon atoms in the SiC substrate, recovering charge neutrality of the overlying graphene.

Detailed analysis of the electron band structure of graphene supports that the substrate is modified by the presence of manganese atoms. Figure 3(a) shows the graphene π band of as-grown graphene (red curves) and graphene with manganese atoms (blue curves) obtained using a Lorentzian fit to the momentum distribution curves (MDCs) for the ARPES intensity maps shown in Figs. 1(a) and 1(c), respectively. To study the change in the dispersion induced by manganese atoms, the dispersions are shown with respect to ED and compared to the calculated π band (black curve) using the local density approximation (LDA) method [27]. While the slope of the dispersion at ED decreases from 0.89 × 106 m/s for as-grown graphene to 0.86 × 106 m/s for graphene with manganese atoms, the latter value is similar to the value for the LDA band of 0.85 × 106 m/s. More importantly, the overall slope of the dispersion of graphene with manganese atoms is lower than that of as-grown graphene, approaching the value for the LDA band.

The slope of the graphene π band is strongly influenced by dielectric environment [28]. When graphene is placed on a metallic substrate, electron-electron interaction is screened by the metallic background from the substrate. Such screening results in the graphene π band convergence towards the LDA band, which describes the case of efficiently screened electron-electron interaction [6,29]. The energy-momentum dispersion of graphene with manganese atoms is in good agreement with the LDA band as shown in Fig. 3(a). This agreement indicates the possibility that the interface of SiC underneath graphene becomes metallic, leading to enhanced screening of electron-electron interaction in graphene compared to that for as-grown graphene. A plausible origin of this screening formation of silicon-manganese bonds, i.e., manganese silicide. Such silicide may not be well ordered, possibly resulting in random site potentials, which is exhibited as the enlarged MDC width shown in Fig. 3(b) (the peak observed at ~0.4 eV below EF and at EF denoted by a black arrow is due to the absence of a well defined quasiparticle state by the energy gap at ED [25]). The formation of a disordered silicon-metal layer in the SiC substrate has been reported to significantly reduce substrate influenc for the cases of epitaxial graphene with gold [13] and tin [15]. The silicon-metal layer could be responsible for the charge transfer back to the substrate, which realizes charge-neutral graphene and further lifting of the sublattice symmetry, leading to a larger energy gap than that of as-grown graphene, as observed in Fig. 1 and which is schematically shown in Fig. 2. However, it is important to note that the understanding of the role of manganese in the formation of gapped nearly free-standing graphene requires additional experimental and theoretical studies. X-ray photoemission spectroscopy will be able to confirm the formation of the silicide layer and model calculations can be conducted to simulate the enhancement of the broken sublattice symmetry.

The interaction of graphene with foreign atoms can induce an interesting spin state in the system. The hybridization between the graphene π band and the cobalt 3d or platinum 5d state results in polarized spin states in graphene [7,30]. In addition, the graphene π band participates in the formation of a spin-dependent many-body ground state in the presence of the cerium 4f state [31]. In fact, when manganese atoms form a cluster, their interatomic spin coupling leads to the RKKY (Ruderman-Kittel-Kasuya-Yosida) type interaction in graphene [32]. Furthermore, the formation of silicide can be boosted in the case of manganese atoms with high chemical reactivity with silicon [33], by performing annealing, that was applied to the samples during the cleaning process before the ARPES measurements. In this way, spin interactions are disturbed, resulting in the absence of magnetic effects. On the other hand, the spontaneous polarization by charge redistribution from silicide [15,34] is assumed to be responsible for the decoupling of graphene from the substrate and the metal-to-insulator transition.

Electronic properties of graphene on an SiC(0001) substrate in the presence of manganese atoms have been investigated using the ARPES technique. The introduction of manganese atoms in the system allows graphene to approach its charge neutrality and to under go a metal-to-insulator transition. In addition, the electron-electron interaction in graphene is sufficiently screened, which is confirmed by the graphene π band agreement with the LDA band, suggesting the possibility that manganese atoms form a metallic silicide layer on top of the SiC substrate. These findings provide not only the experimental evidence of gapped nearly free-standing graphene on an SiC substrate, but also a promising route towards the application of graphene which is semiconducting and charge neutral to electronic devices.

This work was supported by a 2-Year Research Grant from the Pusan National University.

Fig. 1. (a–d) ARPES intensity maps of as-grown graphene (a,b) and graphene with manganese atoms (c,d) taken along and perpendicular to the ΓK direction around the K point as denoted by red lines in the inset. The blue and red arrows denote the electron band structure of single-layer and double-layer graphene, respectively. The black arrow denotes the Dirac energy of single-layer graphene. (e–f) Energy distribution curves for the ARPES data shown in panels (a–d). The black dashed line denotes the manganese 3d state that is non-dispersive. The blue dashed-line in panel (h) denotes the π band of graphene with manganese atoms.
Fig. 2. Cartoons of geometric and electronic structure of graphene before (a) and after (b) the introduction of manganese atoms in the system. Red and blue spheres denote carbon sublattices.
Fig. 3. (a) Energy-momentum dispersions of as-grown graphene (red curve) and graphene with manganese atoms (blue curve) along the ΓK direction of the graphene unit cell denoted by a red line in the inset, compared to the calculated dispersion using the LDA method (black curve). The dispersions are shown with respect to ED. (b) The full with at half maximum of the momentum distribution curves of as-grown graphene (red dots) and graphene with manganese atoms (blue dots).
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# Proof of Homomorphism Property for $(n - k)\mod n$

Pretty easy question probably: How do you prove that $\phi(k)=(n-k)\mod{n}$ satisfies the homomorphism property for the binary structures $\langle\mathbb{Z}_n,+\rangle$ and $\langle\mathbb{Z}_n,+\rangle$?

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What have you tried? –  mixedmath Sep 20 '12 at 0:12
I'm trying to prove it's an isomorphism. What I have now is $\phi(j+k)=(n-(j+k)(\mod{n})=(n-j-k)(\mod{n})$. I don't know how the transition from this to $\phi(j)+\phi(k)$ would be justified. I've proved the 1-1 and onto parts. –  Mike 105 Sep 20 '12 at 0:25
I changed $<\mathbb{Z}_n,+>$ to $\langle\mathbb{Z}_n,+\rangle$. That is standard usage. –  Michael Hardy Sep 20 '12 at 1:41
For general groups, the map $x \mapsto x^{-1}$ is not a homomorphism but for abelian groups it is. This is what you have here, written additively. –  lhf Sep 20 '12 at 2:02
I'm new to LaTeX: How do you do those angle brackets? –  Mike 105 Sep 20 '12 at 3:33

You need to show that for all $a$ and $b$, we have $\phi(a+b)=\phi(a)+\phi(b)$. So, to use number-theoretic terminology, you need to show that if $x\equiv -a\pmod{n}$ and $y\equiv -b\pmod{n}$, then $x+y\equiv -(a+b)\pmod{n}$.

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## Graph theory and combinatoricsproceedings of the cambridge combinatorial conference in honour of paul erdos,cambridge,trinity college,21-25 march,1983 Bollobas, Bela | Academic Press 1984

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## Probabilistic combinatorics and its applicationsproceedings of a symposia held in San Francisco, CaliforniaJanv.14-15 Bollobas, Bela ; Chung, Fan R. K. ; Diaconis, Persi | American Mathematical Society 1991

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Lecture notes in mathematics , 1541

Localisation : Collection 1er étage

branchement à valeur de mesure # calcul stochastique # distribution de Palm # fonctionnelle de Log-Laplace # mesure aléatoire # mesure de Campbell # probabilité # problème de martingale # processus de Markov naissance # processus de Markov à valeur de mesure # processus de construction à valeur de mesure et interaction # regénération # représentation d'amas de Poisson # représentation de De Finetti # retournement # structure de famille # super mouvement Brownien branchement à valeur de mesure # calcul stochastique # distribution de Palm # fonctionnelle de Log-Laplace # mesure aléatoire # mesure de Campbell # probabilité # problème de martingale # processus de Markov naissance # processus de Markov à valeur de mesure # processus de construction à valeur de mesure et interaction # regénération # représentation d'amas de Poisson # représentation de De Finetti # retournement # structure de famille # super ...

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## Expanding graphs :proceedings of a DIMACS workshop#May 11-14 Friedman, Joel | American Mathematical Society 1993

Congrès

- 142 p.
ISBN 978-0-8218-6602-3

DIMACS series in discrete mathematics and theoretical computer science , 0010

Localisation : Collection 1er étage

approche corps de fonction de graphe et diagramme de Ramanuj # construction algébrique de graphe dense de grand contour et # graphe de Cayley aléatoire et expanseur # graphe demi-plan supérieur fini et Ramanujan # graphe en expansion # graphe en expansion hautement tiré de groupe diédral # groupe et expanseur # géométrie spectral et constante de Cheeger # investigation numérique du spectre pour famille de graphe de # laplacien d'hypergraphe # seconde valeur propre et développement linéaire de graphe ré # simulation de chaîne de Markov # échantillonnage uniforme modulo ou groupe de symétrie approche corps de fonction de graphe et diagramme de Ramanuj # construction algébrique de graphe dense de grand contour et # graphe de Cayley aléatoire et expanseur # graphe demi-plan supérieur fini et Ramanujan # graphe en expansion # graphe en expansion hautement tiré de groupe diédral # groupe et expanseur # géométrie spectral et constante de Cheeger # investigation numérique du spectre pour famille de graphe de # laplacien d'hypergraphe # ...

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## Random discrete structuresproceedings of a workshop of the 1993-94 IMA program on emerging applications of probability Aldous, David ; Pemantle, Robin | Springer 1996

Congrès

ISBN 978-0-387-94623-8

The IMA volumes in mathematics and its applications , 0076

Localisation : Colloque 1er étage (MINN)

approximation normale par méthode de Stein # arbre aléatoire # couverture universelle de graphe # distribution aléatoire de masse # distribution de probabilités sur cladogramme # ensemble régénératif # environnement aléatoire # grande déviation # graphe libre de triangle # intersection et limite # marche aléatoire transitoire # matrice positive complètement # méthode du second moment # métrique sur composition et coïncidence # processus aléatoire # recurrence amenabilité # stabilité de processus auto-organisant # structure discrète aléatoire # suite de renouvellement # théorème du cycle impaire long # tresse de jeux de minimax aléatoire # énergie et intersection de chaîne de Markov approximation normale par méthode de Stein # arbre aléatoire # couverture universelle de graphe # distribution aléatoire de masse # distribution de probabilités sur cladogramme # ensemble régénératif # environnement aléatoire # grande déviation # graphe libre de triangle # intersection et limite # marche aléatoire transitoire # matrice positive complètement # méthode du second moment # métrique sur composition et coïncidence # processus ...

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## Combinatorial stochastic processes :école d'été de probabilités de Saint-Flour XXXII#2002 Pitman, Jim ; Picard, Jean | Springer 2006

Congrès

- 256 p.
ISBN 978-3-540-30990-1

Lecture notes in mathematics , 1875

Localisation : Collection 1er étage

arbre aléatoire # mouvement brownien # probabilité combinatoire # processu stochastique # combinatoire asymptotique # position aléatoire

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## Surveys in combinatorics 2011.Papers from the 23rd British combinatorial conferenceExeter # july 3-8, 2011 Chapman, Robin | Cambridge University Press 2011

Congrès

- vii; 437 p.
ISBN 978-1-107-60109-3

London mathematical society lecture note series , 0392

Localisation : Collection 1er étage

combinatoires # graphe topologique # hypergraphe # graphe aléatoire

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## Random graphs'85.Based on lectures presented at the 2nd international seminar on random graphs and probabilistic methods in combinatoricsPoznan # august 5-9, 1985 Karonski, Michal ; Palka, Zbigniew | North-Holland 1987

Congrès

- vii; 354 p.
ISBN 978-0-444-70265-4

North-Holland mathematics studies , 0144

Localisation : Colloque 1er étage (POZN)

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## Séminaire Bourbaki. Vol. 2011/2012:exposés 1043-1058 | Société Mathématique de France 2013

Congrès

- xi; 556 p.
ISBN 978-2-85629-371-3

Astérisque , 0352

Localisation : Périodique 1er étage

Algorithme d'approximation # carte brownienne # cartes planaires # champ libre gaussien # champ moyen # choix social # concentration-compacité # condition nulle # configuration polynomiale # courbe elliptique # D-module holonome # difficulté d'approximation # équation aux dérivées partielles # équations d'Einstein # équations différentielles partielles # équations non-linéaires dispersives # espaces adiques # espaces de Berkovich # espaces homogènes # espaces métriques # espaces normés # espaces perfectoïdes # existence globale # fibré de Higgs # fibré holomorphe plat # forme quartique binaire # formule de KPZ # gravité quantique # groupe de Galois motivique # groupe de Selmer # groupes de Lie # groupes quasi-fuchsiens # hamiltonien # marches aléatoires # mélange exponentiel du fibré des repères # mesures de Liouville # mesures stationnaires # métrique harmonique # modération topologique # monodromie-poids # motifs de Tate mixtes # multizêtas # nonlinéaire # norme d'uniformité # orbites coadjointes # plongement métrique # principe de transfert # programmation semi-définie # Programme de Ribe # progression arithmétique # pureté # rang # réarrangement # Relativité générale # représentations des groupes algébriques réductifs # représentations des groupes de Lie compacts # résonances en espace temps # rigidité # singularités irrégulières # stabilité orbitale # surfaces enfermées # système stellaire auto-gravitant # théorème de Lefschetz difficile # théorie de Hodge # théorie géométrique des invariants # topologie étale # trous noirs # types stablement dominés # variétés de drapeaux # variétés hyperboliques de dimension 3 # Vlasov-Poisson Algorithme d'approximation # carte brownienne # cartes planaires # champ libre gaussien # champ moyen # choix social # concentration-compacité # condition nulle # configuration polynomiale # courbe elliptique # D-module holonome # difficulté d'approximation # équation aux dérivées partielles # équations d'Einstein # équations différentielles partielles # équations non-linéaires dispersives # espaces adiques # espaces de Berkovich # espaces ...

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## Quasi-random graphs and pseudo-random binary sequences Sarközy, Andras | CIRM H

Multi angle

Research talks

The notion of quasi-random graphs was introduced in 1987 by F. R. K. Chung, R. L. Graham and R. M. Wilson, resp. A. Thomason. It has been shown that there is a strong connection between this notion and the pseudorandomness of (finite) binary sequences. This connection can be utilized for constructing large families of quasi-random graphs by considering graphs defined by a circular adjacency matrix whose first column is a binary sequence with strong pseudo-random properties. Starting out from this construction principle one may extend, generalize and sharpen some definitions and results on quasi-randomness of graphs. The notion of quasi-random graphs was introduced in 1987 by F. R. K. Chung, R. L. Graham and R. M. Wilson, resp. A. Thomason. It has been shown that there is a strong connection between this notion and the pseudorandomness of (finite) binary sequences. This connection can be utilized for constructing large families of quasi-random graphs by considering graphs defined by a circular adjacency matrix whose first column is a binary sequence with ...

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## Weighted distances in scale free random graphs Komjathy, Julia | CIRM H

Multi angle

Research talks;Combinatorics;Probability and Statistics

In this talk I will review the recent developments on weighted distances in scale free random graphs as well as highlight key techniques used in the proofs. We consider graph models where the degree distribution follows a power-law such that the empirical variance of the degrees is infinite, such as the configuration model, geometric inhomogeneous random graphs, or scale free percolation. Once the graph is created according to the model definition, we assign i.i.d. positive edge weights to existing edges, and we are interested in the proper scaling and asymptotic distribution of weighted distances.
In the infinite variance degree regime, a dichotomy can be observed in all these graph models: the edge weight distributions form two classes, explosive vs conservative weight distributions. When a distribution falls into the explosive class, typical distances converge in distribution to proper random variables. While, when a distribution falls into the conservative class, distances tend to infinity with the model size, according to a formula that captures the doubly-logarithmic graph distances as well as the precise behaviour of the distribution of edge-weights around the origin. An integrability condition decides into which class a given distribution falls.
This is joint work with Adriaans, Baroni, van der Hofstad, and Lodewijks.
In this talk I will review the recent developments on weighted distances in scale free random graphs as well as highlight key techniques used in the proofs. We consider graph models where the degree distribution follows a power-law such that the empirical variance of the degrees is infinite, such as the configuration model, geometric inhomogeneous random graphs, or scale free percolation. Once the graph is created according to the model ...

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## Random hyperbolic graphs Kiwi, Marcos | CIRM H

Multi angle

Research talks;Combinatorics;Computer Science;Probability and Statistics

Random hyperbolic graphs (RHG) were proposed rather recently (2010) as a model of real-world networks. Informally speaking, they are like random geometric graphs where the underlying metric space has negative curvature (i.e., is hyperbolic). In contrast to other models of complex networks, RHG simultaneously and naturally exhibit characteristics such as sparseness, small diameter, non-negligible clustering coefficient and power law degree distribution. We will give a slow pace introduction to RHG, explain why they have attracted a fair amount of attention and then survey most of what is known about this promising infant model of real-world networks. Random hyperbolic graphs (RHG) were proposed rather recently (2010) as a model of real-world networks. Informally speaking, they are like random geometric graphs where the underlying metric space has negative curvature (i.e., is hyperbolic). In contrast to other models of complex networks, RHG simultaneously and naturally exhibit characteristics such as sparseness, small diameter, non-negligible clustering coefficient and power law degree ...

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## Connected chord diagrams, bridgeless maps, and perturbative quantum field theory Yeats, Karen | CIRM H

Multi angle

Research talks;Combinatorics;Mathematical Physics

Rooted connected chord diagrams can be used to index certain expansions in quantum field theory. There is also a nice bijection between rooted connected chord diagrams and bridgeless maps. I will discuss each of these things as well as how the second sheds light on the first. (Based on work with Nicolas Marie, Markus Hihn, Julien Courtiel, and Noam Zeilberger.)

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## Random cubic planar graphs revisited Rué, Juanjo | CIRM H

Multi angle

Research School;Combinatorics

We analyze random labelled cubic planar graphs according to the uniform distribution. This model was analyzed first by Bodirsky et al. in a paper from 2007. Here we revisit and extend their work. The motivation for this revision is twofold. First, some proofs where incomplete with respect to the singularity analysis and we provide full proofs. Secondly, we obtain new results that considerably strengthen those known before. For instance, we show that the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance, while formerly it was only known that it is linear with high probability.
This is based on a joint work with Marc Noy (UPC) and Clément Requilé (FU Berlin - BMS).
We analyze random labelled cubic planar graphs according to the uniform distribution. This model was analyzed first by Bodirsky et al. in a paper from 2007. Here we revisit and extend their work. The motivation for this revision is twofold. First, some proofs where incomplete with respect to the singularity analysis and we provide full proofs. Secondly, we obtain new results that considerably strengthen those known before. For instance, we show ...

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Ressources Electroniques (Depuis le CIRM)

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text:
# Simple vector subtraction question

1. Aug 31, 2006

### mikefitz

Greeting everyone, I come to you as a lowly/disgruntled undergrad trying to solve a very simple physics problem. This is the only problem I'm having with my assignment, but I cannot seem to make ANY progress with this one!

I've been given two vectors:

http://img266.imageshack.us/img266/4941/picuj0.gif [Broken]

A has magnitude of 5 m and an angle of 55 degrees.
B has a magnitude of 2 m and an angle of 35 degrees.

Here is what I'm supposed to find: Vector C (not shown in the diagram) is the difference of A and B (C = A - B).

What exactly do they mean Vector C? I have no clue what they're talking about - does C begin at the origin?

I don't want the answer given to me, but I would like a sense of direction; thank you so much for your time!

Last edited by a moderator: May 2, 2017
2. Aug 31, 2006

### Saketh

C is a vector that results from subtracting B from A.

By definition, vectors begin at the origin. The endpoint of the vector is its vertex. If I say vector <3, 1>, that means the vector begins at the origin and points to x = 3, y = 1.

You know how to add vectors. You also know what multiplying a vector by -1 does to the vector. So think about subtracting the vector as C = A + (-B).

3. Aug 31, 2006

### mikefitz

Does that mean you are to subtract the magnitude of B from A? - this is the only part of this entire chapter I'm confused on; my book only has a brief write up on the subject and gives no example problems to work on. grr....

4. Aug 31, 2006

### lightgrav

NO! keep track of their directions ...
- B_vector is opposite the B_vector ... points down and to the right.

Add this (Tail-to-tip, like old connect-the-dots) to A,
and C will point from the first tail (at the origin) to the last tip.

5. Sep 1, 2006

### Saketh

When are you adding vectors, you "lift" one of them off of the paper and put its origin point on the arrow point of the other vector. The resulting added vector is the vector that points from the origin to the vertex of the final vector on the chain.

When you are subtracting vectors, you flip one into the opposite direction (due to the negative sign), then add the vectors. The resulting vector is, again, the vector that points from the origin to the final vertex on the chain.

http://www.sparknotes.com/testprep/books/sat2/physics/chapter4section3.rhtml" [Broken]

Last edited by a moderator: May 2, 2017
6. Sep 1, 2006

This may or may not be helpful.

The notation I picked up somewhere helps me remember... (i'll just show by an example).

Lets say you have two vectors $\vec A=(3,0)$ and $\vec B =(0,3)$.

Now lets say you want a vector that points from $\vec B$ to $\vec A$

Call this vector $\vec {BA}$

Then just remember to subtract the second part from the first part, such as:
$\vec {BA} = \vec A - \vec B$

Verify this:

$$\vec {BA} = (3-0,0-3)$$

Now notice that if you put the tail of this vector at the head of $\vec B$ the x component moves over 3, and the y component moves down 3, which puts you at the head of vector A.

Side note:
I don't know why the vector symbols is not going accross the entire BA. I don't have time to look up the proper LaTeX for it. But just draw it yourself when doing the problems.

Last edited: Sep 1, 2006
7. Sep 1, 2006

### Staff: Mentor

As FrogPad is suggesting, it is much more practical to add and subtract the vectors' components in rectangular notation. Do you know how to convert the polar notation that you were given (length and angle) to rectangular notation (deltaX, deltaY)?


text:
# Is it reflexive: everyone who has visited Web page a has also visited Web page b, for all webpages

This is a question from "Discrete Mathematics and Its Applications":

Determine whether the relation R on the set of all Web
pages is reflexive, symmetric, antisymmetric, and/or tran-sitive,
where(a, b) ∈ R if and only if

a) everyone who has visited Web page a has also visited Web page b .

a) Reflexive, transitive

I don't understand well why it's "Reflexive". If there is a web page which has never been visited by anybody, (a,a) won't be belonged to R, that R should not be reflexive, right?

Do I misunderstand something wrong?

Thanks for @Brian M. Scott's answer, I understand it now. Let me add something important here:

The description of $R$ is:

$R$={$(x,y) | \forall p (F(p,x) \to F(p,y))$}, where $F(p,x)$ means "p has visited x", and $p \in P$ where P is "all people".

When page $x$ has never been visited, that $F(p,x), p \in P$ is always false, that $\forall p (F(p,x) \to F(p,y))$ is always true, that F(x, any y) is always true.

That's why the relation is "Reflexive".

-
It will be helpful for you to review the definition of a relation first. What does $aRb$ mean? What does $aba$ mean? – Calvin Lin Jan 7 '13 at 4:11
"Everyone who has visited the Sun has (also) visited the Sun" is true. – André Nicolas Jan 7 '13 at 4:16

If $a$ is a web page, let $V(a)$ be the set of people who have visited $a$. Then $\langle a,b\rangle\in R$ if and only if $V(a)\subseteq V(b)$. Now it’s always true that $V(a)\subseteq V(a)$, even if $V(a)=\varnothing$, so it’s always true that $\langle a,a\rangle\in R$. Thus, $R$ really is reflexive.
Here’s a slightly different way to look at it. If $a$ and $b$ are particular web pages, how could you prove that $\langle a,b\rangle\notin R$? You would have to find someone who has visited $a$ but has not visited $b$. If no one has visited $a$, you can’t do this: you can’t find anyone who has visited $a$, let alone someone who has visited $a$ but not $b$! This shows that if $a$ has never been visited, then $\langle a,b\rangle\in R$ for all $b$ (and hence certainly for the particular case $b=a$).
$R$={$(x,y) | \forall p (F(p,x) \to F(p,y))$}, where $F(p,x)$ means "p has visited x". Is it right? – Freewind Jan 7 '13 at 4:39
@Freewind: Yes, that’s one correct way to describe it, though it would be better if you wrote $\forall p\in P$ and defined $P$ to be the set of all people. – Brian M. Scott Jan 7 '13 at 4:41
If page $x$ has never been visited, that $F(p,x)$ will always be false, that $\forall p (F(p,x) \to F(p,y))$ will be true, so $(x, any y)$ belong to $R$. – Freewind Jan 7 '13 at 4:41


text:
# Circumference of a circle in parametrics

1. Feb 10, 2010

### hangainlover

1. The problem statement, all variables and given/known data

x^2+y^2=1

2. Relevant equations

length of curve square root (1+(dx/dy)^2)dy or square root ((dx/dt)^2 + (dy/dt)^2) dt

3. The attempt at a solution[/b

I isolated y and got y= square root (1-x^2)

finding dy/dx = -x/square root (1-x^2)
i plugged that into the formula but i did not get the correct answer.

im aware that if i do integral of square root (1+ (dy/dx)^2 ) i have to do imporper integral as i get undefined for denominator at x=-1,1 so what should i do
we all know that answer should be 2pi

2. Feb 10, 2010

### Dick

Ignore the 'undefined' issue. Just do a trig substitution to solve the integral. Once you've done the trig substitution you won't see the singularity. BTW you'll get pi, not 2*pi. Your parametrization only covers half the circle.

3. Feb 10, 2010

### hangainlover

I know, i can get the answer by defining the circle in trig.

But, shouldn't i be able to do it in conventional way and by paramatrization as well?
i just need to take care of the x values and t values where the function gets undefined.
My teacher wants me to do it in these two ways.

4. Feb 10, 2010

### Dick

Even to do it in the x-y way you need to integrate 1/sqrt(1-x^2) from -1 to 1, right? Even if you haven't parametrized it as a trig function you still need to do a trig substitution to solve that integral. That's what I'm talking about.


text:
## what are polynomials and write their degree of 2×2 + 3/x -11

Question

what are polynomials and write their degree of 2×2

+ 3/x -11

in progress 0
2 months 2021-08-13T14:30:46+00:00 1 Answer 0 views 0


text:
# Alternating Series and P-series convergence

1. Jul 28, 2008

### kylera

Alternating Series and P-series "convergence"

I couldn't resist trying out a pun. Anyway, onto the question:

1. The problem statement, all variables and given/known data
Test the series for convergence/divergence:
$$\sum^{\infty}_{n=1}\frac{-1^{n-1}}{\sqrt{n}}$$

2. Relevant equations
Alternating Series Test and possibly p-series test...

3. The attempt at a solution
The expression for an alternating series goes as a(n) = (-1)^(n-1) * b(n). Having said that, it's obvious that b(n) is

$$\frac{1}{\sqrt{n}}$$

which can be re-written as $$\frac{1}{n^\frac{1}{2}}$$. But by the rules of the p-series, since 0.5 is obviously lesser than 1, I came to the conclusion that the series diverges. Instead, the answer states that the series does indeed converge. Can anyone help shed some light in this?

2. Jul 28, 2008

### rock.freak667

Re: Alternating Series and P-series "convergence"

I think for the p-series test, the numerator must be '1' only.

3. Jul 28, 2008

### Dick

Re: Alternating Series and P-series "convergence"

Forget the p-series test. Concentrate on the alternating series test. b(n) is decreasing towards 0. Sorry, I don't get the 'pun'. Am I being thick?

4. Jul 28, 2008

### d_leet

Re: Alternating Series and P-series "convergence"

I don't think you are since I don't get it either.

5. Jul 28, 2008

### rock.freak667

Re: Alternating Series and P-series "convergence"

Well usually for the p-series test, the only examples with fractions I've ever seen is where the function is in the form

$$\frac{1}{f(n)}$$

6. Jul 28, 2008

### Dick

Re: Alternating Series and P-series "convergence"

Sure, in fact, it applies only to series of the form 1/n^p or n^(p). I wasn't saying you were wrong. It's useful here if you want to discuss absolute convergence. It is useless for the given function.

7. Jul 29, 2008

### HallsofIvy

Staff Emeritus
Re: Alternating Series and P-series "convergence"

The "p-series" only applies to positive series. That's why Dick says it is useful to discuss absolute convergence. If your series has both positive and negative terms then it may converge "conditionally". It will converge absolutely only if the series of absolute values converges.

By the "alternating series test", the series
$$\sum \frac{(-1)^n}{\sqrt{n}}$$
converges.

By the "p- series test"
$$\sum \left|\frac{(-1)^n}{\sqrt{n}}\right|= \sum \frac{1}{\sqrt{n}}= \sum \frac{1}{n^{1/2}}$$
does NOT converge and so the original series converges conditionally, not absolutely.


text:
# Three cats

If three cats eat three mice in three minutes, after which time 260 cats eat 260 mice?

Result

t =  3 min

#### Solution:

$t=3 \ \text{min}$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Showing 1 comment:
Math student
this is so confusing.

Tips to related online calculators
Do you want to convert time units like minutes to seconds?

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text:
# Julia output not displaying in quarto

Hi everyone,

I am writing a quarto document, which includes code chunks in R, Python and Julia.

The output from the Python and R chunks is displaying correctly. However, while the code for the Julia chunks is displaying, the output is not - see the below image:

My YAML header is included below:

format:
html:
theme: spacelab
html-math-mathod: mathjax
toc: true
toc-location: right
number-sections: true
code-copy: true
code-fold: true
code-tools: true
monofont: Fira Code
execute:
message: false
warning: false
bibliography: references.bib

Please advise on how I can see the output of my Julia code chunks in this document too.

Many thanks,
Dom

2 Likes

Can you share the content of the .qmd file ? How does the Julia chunk looks like ?

thanks


text:
# How to show that $f$ is an odd function?

An entire function $f$ takes real $z$ to real and purely imaginary to purely imaginary. We need to show that $f$ is an odd function. well, $f=\sum_{n=0}^{\infty}a_nz^n$ what I can say is $f(\mathbb{R})\subseteq\mathbb{R}$ and $f(\mathbb{iR})\subseteq\mathbb{iR}$

How to proceed, please give me hint.

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ahhh! I just got it since $f(\mathbb{R})\subseteq\mathbb{R}$ so each $a_n$ is real again since $f(\mathbb{iR})\subseteq\mathbb{iR}$ each $a_n=0$ for even $n$ am I right? – La Belle Noiseuse Jun 14 '12 at 9:08
You may be expected to show that the two conditions respectively force (i) the $a_n$ to be real and (ii) the even coefficients to be $0$. – André Nicolas Jun 14 '12 at 9:28
@Mex that is correct. – TenaliRaman Jun 14 '12 at 9:30
You may also use Schwarz Reflection Principle and show $f(-z) = -f(z)$. – TenaliRaman Jun 14 '12 at 9:32
@Mex: I encourage you to answer your own question and accept it, if you think you have the correct answer. – mixedmath Jun 14 '12 at 10:38

## 1 Answer

I am writing, enlarging and enhancing (hopefully...) Mex's answer to his own question (kudos!) and I'll be happy to erase this answer of mine if he decides to write down his.

We have that $$f(z)=\sum_{n=0}^\infty a_nz^n$$ because $\,f\,$ is entire, and by the given conditions we have$$(1)\,\,f(r)=\sum_{n=0}^\infty a_nr^n\in\mathbb{R}\,,\,\,\,r\in\mathbb{R}$$$$(2)\,\,f(ir)=\sum_{n=0}^\infty a_n(ir)^n\in i\mathbb{R}\,\,,\,\,r\in\mathbb{R}$$but we have that $$\sum_{n=0}^\infty a_n(ir)^n=\sum_{n=0}^\infty i^n (a_nr^n)=\sum_{n=0}^\infty (-1)^na_{2n}r^{2n}+i\sum_{n=0}^\infty (-1)^na_{2n+1}r^{2n+1}$$and as the above is purely imaginary we get that $\,a_{2n}=0\,,\,\forall n\in\mathbb{N}\,$ , so the power series of the function has zero coefficients for the even powers of $\,z\,$ and is thus a sum of odd powers and trivially then an odd function.

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Don :) my great pleasure, thank you very much. – La Belle Noiseuse Jun 14 '12 at 14:42
(1)You've very much welcome, @Mex, and (2) thanks for giving the solution to your own question. – DonAntonio Jun 14 '12 at 14:50


text:
# [✓] Locator constrained by polygon and NSolve

GROUPS:
 Dear Community,I have a polygon with a constrained locator in it. I would like to draw a horizontal line from the locator towards the left boundary of the polygon, which is also given by a stewise function called qgHSZ. I try to achieve this with NSolve, but I get some strange warnings, like "Part 2 of {0.5,10.} does not exist" Why?? "NSolve was unable to solve the system with inexact coeficients, etc." Not clear either. If I test NSolve below the plot, it works fine.What do I do wrong? Notebook attached. Tx for the kind help in advance, regards, Andras Attachments:
 This seems to work: DynamicModule[{pt = {0.5, qgHSZ[.2]}, from = .2, x}, {Graphics[{Dynamic[Line[{{from , pt[[2]]}, pt}]], LightBlue, Opacity[0.9], Polygon[pts], Locator[ Dynamic[pt , (pt = RegionNearest[Polygon[pts], #]; from = x /. NSolve[ qgHSZ[x] == pt[[2]], x][[1]]) &]]}, Axes -> True, AspectRatio -> 1], Dynamic[pt]}]


text:
# I have this linear PDE…

Hi,

The PDE in question is: $A P_{yy}(y,z) + B P_{zz}(y,z) + ( [ C y -D z] P(y,z) )_y + ( [ D y + C z ] P(y,z) )_z=0,$

where subscript $y,z$ indicates derivatives and $A,B,C,D$ are real. The PDE is the stationary bit of a Fokker-Planck equation derived from a generally complex Langevin equation, i.e. $x = y+iz$.

I was wondering if anyone knew of a standard table/textbook that details the solution or how to solve PDEs like the above? With no complex part ($z=0$) the solution is a Gaussian (with suitable restrictions on the coefficients).

It is possible to let $A$ or $B$ be zero and the ensuing equation is still meaningful.

Thanks

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## 3 Answers

If you are looking for solutions in the whole plane, take the Fourier transform. You get a first order PDE which you can then analyze by the method of characteristics.

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This is just a comment on Rafa's answer, but I don't seem to find the way to make comments. The PDE is not necessarily elliptic (take e.g. A = 1 and B = -1, resulting in the wave equation). If A and B are (identically) zero then the equation just says that the divergence of the vector field $Y(y, z)$ with entries $Y_1(y,z) = (Cy - Dz)P(y, z)$ and $Y_2(y, z) = (Dy + Cz)P(y, z)$ is zero. If the domain is simply connected, then that is equivalent to saying that there exists a scalar function (a potential) $V$ such that $Y = \nabla^{\perp} V$.

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Ooo, that's right, I agree –  guacho May 30 '13 at 8:36

This is an elliptic operator. I think that Chapter 6 of Evans's book "Partial differential equations" is appropriate if the domain is bounded. In the whole plane you can use Fourier techniques, isn't it?

I hope this help you

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text:
A Explicit kernel formulas

# The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points\tnotereffund

## Abstract

We study the adjoint of the double layer potential associated with the Laplacian (the adjoint of the Neumann–Poincaré operator), as a map on the boundary surface of a domain in with conical points. The spectrum of this operator directly reflects the well-posedness of related transmission problems across . In particular, if the domain is understood as an inclusion with complex permittivity , embedded in a background medium with unit permittivity, then the polarizability tensor of the domain is well-defined when belongs to the resolvent set in energy norm. We study surfaces that have a finite number of conical points featuring rotational symmetry. On the energy space, we show that the essential spectrum consists of an interval. On , i.e. for square-integrable boundary data, we show that the essential spectrum consists of a countable union of curves, outside of which the Fredholm index can be computed as a winding number with respect to the essential spectrum. We provide explicit formulas, depending on the opening angles of the conical points. We reinforce our study with very precise numerical experiments, computing the energy space spectrum and the spectral measures of the polarizability tensor in two different examples. Our results indicate that the densities of the spectral measures may approach zero extremely rapidly in the continuous part of the energy space spectrum.

Résumé

Nous étudions l’adjoint du potentiel de double couche associé à l’opérateur de Laplace (l’adjoint de l’opérateur de Neumann-Poincaré) défini sur la frontière d’un domaine de contenant des points coniques. Le spectre de cet opérateur est intimement lié à la résolution de problèmes de transmission à travers . En particulier, dans le contexte de la propagation des ondes électromagnétiques, si le domaine délimité par représente une inclusion contenant un matériau de permittivité complexe , immergé dans un milieu infini de permittivité égale à 1, on peut définir le tenseur de polarisabilité dès que le rapport appartient à l’ensemble résolvent de l’opérateur au sens de la norme d’énergie. Nous étudions des surfaces qui possèdent un nombre fini de points coniques à symétrie de rotation. Lorsque l’opérateur est défini sur l’espace d’énergie, nous montrons que son spectre essentiel est un intervalle. Lorsqu’il est défini dans l’espace , i.e. pour des fonctions de carré intégrable sur , nous montrons que son spectre est constitué d’une union de courbes, en dehors desquelles on peut calculer l’indice de Fredholm de l’opérateur, comme l’indice par rapport à ces courbes. Nous donnons des formules explicites, en fonction de l’angle d’ouverture des points coniques. Nous complétons notre étude par des expériences numériques très précises, où, pour deux exemples, nous calculons le spectre de l’opérateur au sens de l’espace d’énergie et les mesures spectrales du tenseur de polarisabilité. Nos résultats suggèrent que les densités des mesures spectrales approchent zéro extrêmement rapidement dans la partie continue du spectre au sens de l’espace d’énergie.

###### keywords:
layer potential, Neumann–Poincaré operator, spectrum, polarizability
###### Msc:
[2010] 31B10, 45B05, 45E05
1\tnotetext

[fund]This work was supported by the Swedish Research Council under contract 621-2014-5159.

\cortext

[cor1]Corresponding author

## 1 Introduction

Let be a connected Lipschitz surface, enclosing a bounded open domain and with surface measure . We are interested in the spectrum of the layer potential operator

 KΓf(r)=12π∫ΓKΓ(r,r′)f(r′)dσ(r′),r∈Γ, (1)

based on the normal derivative of the Newtonian kernel

 KΓ(r,r′)=⟨r−r′,νr⟩|r′−r|3, (2)

where denotes the outward unit normal of . may also be considered for planar Lipschitz curves , in which case the kernel is given by .

Knowledge about the spectrum of leads to existence and uniqueness results for boundary value problems involving the Laplacian on the interior and exterior domains and of . For example, layer potential techniques may be used to solve the classical Dirichlet and Neumann problems for by understanding the Fredholm theory of and , respectively (51).

When is non-smooth, for example if has corners in 2D, or edges or conical points in 3D, the spectrum of is highly dependent on the space . For example, suppose that is a curvilinear polygon in the plane. is always invertible (51) when is Lipschitz, but in the polygonal case there always exist , depending on the opening angles of the corners of , such that is not Fredholm (34); (46). The underlying explanation for this is that when is an infinite wedge, the model domain to analyze corners; then, by homogeneity of its kernel, may be realized as a block matrix of Mellin convolution operators. These convolution kernels depend on , accounting for the dependence on of the spectrum (18). In 3D, similar results were shown in (19) in the idealized cases of being an infinite straight cone or an infinite three-dimensional wedge. We refer also to (40) for an extensive account of the -theory in 2D, although with results only stated for .

In this paper we will, for surfaces , consider the action of on two different spaces: and the energy space . The energy space consists of the distributions on whose single layer potentials have finite energy in . It is identifiable with the Sobolev space of index on the boundary. The energy space stands out as the most natural space on which to consider for many reasons, one of them being that is self-adjoint and therefore, in contrast with the -theory, has a real spectrum.

Our interest in the entire spectrum of arises from the transmission problem

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩∫R3|∇U|2dV<∞,ΔU(r)=0,r∈R3∖Γ,TrintU(r)=TrextU(r),r∈Γ,∂extνU(r)=ϵ∂intνU(r)−g(r),r∈Γ. (3)

Here , and and denote the boundary trace and normal derivative of interior approach, and the corresponding operators of exterior approach. If , it turns out that there exists a solution of (3) satisfying if and only if there is such that

 (KΓ−z)f=g1−ϵ,z=−1+ϵ1−ϵ.

In the special case that for a vector , then solving the transmission problem is involved in computing the polarizability tensor (10); (20); (28); (47) of . In this setting, the domain is an inclusion with complex permittivity in an infinite space of permittivity 1. The polarizability tensor is associated with a set of spectral measures that arise from the spectral measure of , see Section 2.2. Atoms in these spectral measures correspond to values of for which surface plasmon resonances can be excited (2); (3). However, not every eigenvalue of necessarily produces a singularity in the polarizability tensor; we call such eigenvalues dark plasmons. In Section 7.3 we will observe an abundance of dark spectra for the type of surface that we will consider. More precisely, the described relationship between the transmission problem (3) and plasmonic resonances holds in the quasi-static approximation of the Maxwell equations. In the setting of smooth surfaces , detailed analysis of plasmonic resonances using the full Maxwell equations and justification of the quasi-static approximation can be found in (1); (4). Note that the spectrum of is pure point when is smooth.

For a plane polygon and , , the spectrum of (a more general version of) the transmission problem (3) was studied in (14). In (31), the spectral resolution of was determined in a model case where is constructed from two intersecting circles (equivalent to the infinite wedge). For a general curvilinear polygon in 2D, the essential spectrum of was determined in (44).

###### Theorem 1.1 ((44)).

Suppose that is a curvilinear polygon with corners of angles . Then the spectrum of consists of an interval and a sequence of eigenvalues with no limit point outside the interval,

 σ(KΓ,E)={x∈R:|x|≤max1≤j≤N∣∣∣1−αjπ∣∣∣}∪{λk}.

See also (24) for a numerical experiments in agreement with this theorem. In the special case that coincides with two line segments in a neighborhood of each corner, a different approach to Theorem 1.1, yielding more information, very recently appeared in (6). In three dimensions, only a few results concerning the entire spectrum seem to be available. As mentioned, the -theory for infinite straight cones and wedges was considered in (19). Also for the infinite straight cone, the generalized eigensolutions to the transmission problem (3) were explicitly computed in (32); (41); (48) – these will be important in our determination of the spectrum of . For the more general type of infinite cone , where is a smooth curve on the sphere, the invertibility of on certain weighted Sobolev spaces has via Mellin convolutions been reduced to the invertibility of a parametric system of operators on (9); (45).

In the current contribution, we will characterize the spectrum of and in the case that is a rotationally symmetric surface with a conical point, see Figure 1. Our main results straightforwardly generalize to surfaces with a finite number of conical points, each of which is locally rotationally symmetric around some axis. However, since the level of complexity is already quite high, we will never do so explicitly.

We now state our main theorems, beginning with our result on the -spectrum. In the statement, denotes an associated Legendre function of the first kind (see the Appendix), and denotes its derivative in .

###### Theorem 5.28.

Let be a closed surface of revolution with a conical point of opening angle , obtained by revolving a -curve . For , denote by the closed curve

 Πn=⎧⎨⎩Pniξ(cosα)˙Pniξ(−cosα)−Pniξ(−cosα)˙Pniξ(cosα)Pniξ(−cosα)˙Pniξ(cosα)+Pniξ(cosα)˙Pniξ(−cosα):−∞≤ξ≤∞⎫⎬⎭,

with orientation given by the -variable. Then the operator has essential spectrum

 σess(KΓ,L2)=∞⋃n=−∞Πn.

If , then has Fredholm index

 index(KΓ−z)=∞∑n=−∞W(z,Πn)=W(z,Π0)+2∞∑n=1W(z,Πn)

where denotes the winding number of with respect to and the right-hand side is always a finite sum. In particular, every point lying inside one of the curves belongs to the spectrum .

Whenever is not a real number, it holds that , so that

 index(KΓ−z)=−codimranKΓ,z∈C∖R.

In particular, if (so that lies outside every curve ), then either is invertible or is real and an eigenvalue of .

Theorem 5.28 is illustrated in Figure 2. After reversing the signs of the winding numbers, the first paragraph of the theorem applies equally well to the adjoint operator , known as the Neumann–Poincaré operator. As a consequence, the number of eigenfunctions of to the eigenvalue is equal to the winding number of with respect to the essential spectrum, except at certain exceptional real values .

Next, we state our characterization of the -spectrum.

###### Theorem 6.33.

Let be a closed surface of revolution with a conical point of opening angle , obtained by revolving a -curve . For , denote by the closed interval

 Σn=⎧⎪⎨⎪⎩Pniξ−1/2(cosα)˙Pniξ−1/2(−cosα)−Pniξ−1/2(−cosα)˙Pniξ−1/2(cosα)Pniξ−1/2(−cosα)˙Pniξ−1/2(cosα)+Pniξ−1/2(cosα)˙Pniξ−1/2(−cosα):−∞≤ξ≤∞⎫⎪⎬⎪⎭.

Then the self-adjoint operator , where is the energy space of , has essential spectrum

 σess(KΓ,E)=∞⋃n=−∞Σn.

Hence, the spectrum of consists of this interval and a sequence of real eigenvalues with no limit point outside of it,

 σ(KΓ,E)={λk}∪σess(KΓ,E).

In Section 7 we will develop a method to numerically determine the polarizability tensor and spectrum of for rotationally symmetric surfaces . We offer one of our numerical results already here, which at the same time illustrates Theorem 6.33. In the proof of Theorem 6.33 we decompose according to its Fourier modes, . Figure 3 demonstrates the indicator function for mode , which detects the spectrum of , for a surface of opening angle . The set where the indicator function is equal to coincides with the interval of Theorem 6.33, i.e. the essential spectrum of . The points where the indicator function is correspond to eigenvalues. It turns out (only numerically demonstrated) that in this case there is an infinite sequence of eigenvalues outside the essential spectrum, and every eigenvalue but one yields a plasmon resonance.

We now explain the layout of the paper, with some remarks on the content of each section. Section 2 contains preliminary material on layer potentials, the energy space, the transmission problem, limit polarizability, Fredholm operators, Mellin transforms, Sobolev spaces and singular integral operators.

In Section 3 we study the model case in which is an infinite straight cone. We provide the spectral resolution of both operators and . The first case is quite straightforward, and the relevant analysis appears implicitly in (19). Each modal operator is in this case unitarily equivalent to a Mellin convolution operator, and this leads to the spectral resolution. On the energy space we make use of a special norm related to the single layer potential which has several advantages. For one, this norm decomposes orthogonally with respect to the Fourier modes. Secondly, it allows us to exploit that we can calculate the action of the single layer potential on the generalized eigenfunctions of on the infinite cone. We remark that for the case of intersecting disks, the same norm was used in (31) to realize the spectral theorem of .

In Section 4 we show, in a certain sense, that is a compact perturbation of , where is a straight cone of the same opening angle. The proof proceeds by writing the difference of kernels as a sum of products of Riesz kernels with smooth, small kernels. The Riesz transforms are however not bounded on ; since this would contradict ((39), Eq. (6.50)). Hence, the indicated argument provides compactness on , but for we have to work harder. We combine certain algebraic identities (the Plemelj formula) with further estimates and real interpolation in this case.

In Section 5 we prove Theorem 5.28. The index formula is proven by showing that the modal operators are pseudodifferential operators of Mellin type, for which there is a well-developed symbolic calculus (17); (35); (36). Theorem 6.33 is proven in Section 6. The method of proof is to first show that on the infinite cone, the singularities of at the origin and at infinity contribute equally to the essential spectrum. The theorem is then pieced together from the results in Sections 3 and 4.

Section 7 contains our numerical results. We first define the indicator function, and establish its properties. Then we give an overview of the numerical method, after which we present numerical results on the polarizability tensor and the spectrum of for the two surfaces illustrated in Figure 1.

Finally, the Appendix contains explicit expressions for the various modal kernels we will consider, in terms of special functions. Our theory and numerical method both depend on these explicit formulas. In particular, we will defer the proof of the technical Lemma 3.10 to the Appendix.

#### Notation

If and are two non-negative quantities depending on some variables, we write to signify that there is a constant such that . If and , we write .

## 2 Background, definitions and notation

### 2.1 Single and double layer potentials

For a function on , its single layer potential is given by

 SΓf(r)=∫ΓSΓ(r,r′)f(r′)dσ(r′),r∈R3, (4)

where

 SΓ(r,r′)=12π1|r−r′|.

Note that is a harmonic function for . If is any reasonable function or distribution, will have traces from both the interior domain and the exterior domain . Due to the weak singularity of the kernel, these traces coincide with evaluating directly on ,

 TrintSΓf(r)=TrextSΓf(r)=SΓf(r),r∈Γ.

In the most general case, these traces may be understood in the sense of convergence in nontangential cones for almost every point of (51), or in a distributional sense (43). Most of the time we will consider as map directly on , since the well-posedness of the interior and exterior Dirichlet problems ensure that can be uniquely identified with its values on , see (43).

The layer potential , evaluated on the boundary, is given by the principal value integral (1) with kernel defined in (2). The adjoint operator (with respect to ) is usually referred to as the boundary double layer potential, or the Neumann–Poincaré operator. Note also that the choice of normalizing constant in front of (1) may be different in other works.

We will consider two different function/distribution spaces for the action of . First, we will consider as an operator on . is always bounded on (51), but note that is not a self-adjoint operator in this space. When has singularities, so that is not a compact operator, the spectrum of on is typically not real. This is illustrated by our main theorem on the -spectrum, Theorem 5.28. See also (40) for the 2D-case.

The second space we will consider is the Hilbert space , obtained by completing in the positive definite scalar product

 ⟨f,g⟩E=⟨SΓf,g⟩L2(Γ).

By applying the classical jump formulas for the interior and exterior normal derivatives of and Green’s formula, we have that

 ⟨f,f⟩E=12∫R3|∇SΓf|2dV. (5)

A proof, which carries over verbatim to the Lipschitz case (see (43)) may be found in ((33), Lemma 1). Here denotes the usual volume element on . Hence, we refer to as the energy space, as it consists of charges generating single layer potentials with finite energy in . In light of this physical interpretation, it is not a complete surprise that is self-adjoint as an operator on . Indeed, from the Plemelj formula

 SΓKΓf(r)=(KΓ)∗SΓf(r),r∈Γ (6)

it follows that

 ⟨KΓf,g⟩E=⟨f,KΓg⟩E,

see (31); (33); (43). By considering trace theorems and well-posedness of Dirichlet problems, it can be deduced that the -norm is equivalent to the Sobolev norm of index on the boundary (see Section 2.5),

 ∥f∥E≃∥f∥H−1/2(Γ). (7)

Again we refer the reader to (33), or to (43) for a treatment explicitly for the Lipschitz case. By interpolating between and , where is bounded by the classical theory (51), it now follows that is bounded.

It is known ((11), Theorem 2.5) that as an operator on the spectrum of is contained in ,

 σ(KΓ,E)⊂(−1,1]. (8)

However, without additional hypotheses on such as convexity or smoothness, it is not even known if the essential norm of is less than ,

 ∥KΓ∥L2(Γ)→L2(Γ),ess≤1?

We refer to (52) for a discussion.

In addition to bounded domains, we will consider one instance of an unbounded surface. Namely, we will consider an infinite straight cone of opening angle , , . In general the layer potential theory for domains with non-compact boundary is rather delicate, but in our particular case is a Lipschitz graph. In any case, since will be our model for studying domains with axially symmetric conical points, we will make precise calculations from which the boundedness and other basic properties of and will be apparent. All of the properties of , , and mentioned in this subsection continue to hold, except that we (the authors) are not entirely sure about the available results on the Dirichlet problem. In particular, we are not sure if (7) holds. However, in view of (5) and the boundedness of the trace (38), we at least have that

 ∥f∥E=sup∥g∥E=1⟨SΓαg,f⟩L2(Γα)≤sup∥g∥E=1∥TrSΓαg∥H1/2(Γα)∥f∥H−1/2(Γα)≲∥f∥H−1/2(Γα). (9)

Furthermore, if is a smooth compactly supported function, then

 ∥ρf∥E≃∥ρf∥H−1/2(Γα), (10)

with implicit constants possibly depending on .

### 2.2 The transmission problem and limit polarizability

In the transmission problem (3), with , the normal derivatives and of exterior and interior approach need to be understood in a distributional sense. Making the ansatz , the jump formulas

 ∂intνSΓf(r)=f(r)−KΓf(r),∂extνSΓf(r)=−f(r)−KΓf(r),r∈Γ (11)

imply that solves the transmission problem if and only if and

 (KΓ−z)f=g1−ϵ,z=−1+ϵ1−ϵ.

In fact, in the case that is a bounded surface, any solution to the transmission problem which satisfies must be of this form, as mentioned in the introduction. See ((28), Proposition 5.1).

To define the polarizability tensor of we understand as an inclusion with permittivity , embedded in infinite space of permittivity . For a unit field , we seek a potential such that

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩∫R3|∇U−e|2dV<∞,ΔU(r)=0,r∈R3∖Γ,TrintU(r)=TrextU(r),r∈Γ,∂extνU(r)=ϵ∂intνU(r),r∈Γ.

The single layer potential ansatz

 U(r)=e⋅r+SΓρ(r)

yields ((28), Section 2) the equation

 (KΓ−z)ρ=ge,ge(r)=(e⋅νr),z=−1+ϵ1−ϵ.

If the solution exists uniquely for all , then the polarizability tensor , a linear map on , scaled by the volume of , is defined by

 ω(z)e=(ϵ−1)|int(Γ)|∫int(Γ)∇U(r)dV(r).

To evaluate the polarizability, we make use of Green’s formula

 ∫int(Γ)⟨∇U,∇V⟩dV=∫ΓU¯¯¯¯¯¯¯¯¯¯¯¯¯∂intνVdσ, (12)

valid for and harmonic in and of sufficient smoothness.

We suppose now that is rotationally symmetric around the -axis. Then is diagonal, and its first two diagonal entries are equal, . We refer to as polarizability in the -direction, . Applying (12) and the jump formulas (11) yields that

 ωjj(z)=ej⋅ω(z)ej=∫Γρ(r)hej(r)dσ(r)=⟨(KΓ−z)−1gej(r),hej(r)⟩L2(Γ), (13)

where denotes the th unit vector in the standard basis of , and

 he(r)=−2e⋅r|int(Γ)|.

In Section 7.1 we will see that (13) is associated with a spectral measure ,

 ωjj(z)=∫1−1dμj(s)s−z. (14)

This statement is a little more subtle than it appears, since is not a self-adjoint operator in the -pairing. An appropriate formalism was developed in ((28), Section 5), and we shall carry out the corresponding details for our situation in Section 7.1. Alternative approaches may be found in (10); (20); (21); (42). Some of these references concern the effective permittivity tensor rather than polarizability. However, the polarizability tensor may be viewed as a limiting case of the effective permittivity tensor.

By the representation of the polarizability as a Cauchy integral (14), the limit

 ω−jj(x)=limy→0−ωjj(x+iy)

exists almost everywhere , even when lies in the support of . We refer to as the limit polarizability. When lies outside the support of the limit polarizability and polarizability coincide. For axially symmetric domains with a conical point, we will find that the spectral measure typically has an absolutely continuous part, in addition to a possible singular part. The absolutely continuous part is recognized by the fact that almost everywhere it holds that

 μ′j(x)=−Imω−jj(x)π.

By ((28), Remark 5.1 and Theorem 5.2), is a positive measure, and and satisfy that

 ∫Γρ(r)dσ(r)=0, (15)
 ∫1−1dμj(s)=−2. (16)

Let be the pure point part of . By ((28), Theorem 5.6) there are eigenvectors and of and , respectively, normalized so that

 ⟨ϕi,ψk⟩L2(Γ)=δik,

such that

 ∫Γdμppj(s)=∑iuivi,ui=⟨ϕi,hej⟩L2(Γ),vi=⟨ψi,gej⟩L2(Γ).

In particular, if has no singular continuous part, then (16) takes the form

 ∑iuivi+∫1−1μ′j(x)dx=−2. (17)

We strongly believe, but will not prove, that never has a singular continuous part for the surfaces we consider. We shall use the rules (15) and (17) to verify the accuracy of our numerical results.

### 2.3 Fredholm operators

Recall that a bounded operator on a Hilbert space is Fredholm if it has closed range and both its kernel and cokernel are finite-dimensional. Equivalently, is Fredholm if and only if it is invertible modulo compact operators. If is Fredholm, its index is given by

 indexS=dimkerS−codimranS.
###### Definition 2.2.

If two operators and on Hilbert spaces and are unitarily equivalent, we write that .

###### Definition 2.3.

We write if there exist Hilbert spaces and and a compact operator such that is similar to .

The point of the above definition is that if and , it holds that is Fredholm if and only if is Fredholm and then the Fredholm indices satisfy

 index(S−z)=index(T−z).

For a (not necessarily self-adjoint) operator we will denote its essential spectrum in the sense of invertibility modulo compacts by .

###### Definition 2.4.

The essential spectrum of is the set

 σess(S,H1)={z∈C:K−z%isnotFredholm}.

We will also make use of another concept of essential spectrum, also invariant under compact perturbations. We say that a bounded sequence is a singular sequence for the operator and spectral point if has no convergent subsequences and in as .

###### Definition 2.5.

The point belongs to if and only if there is a singular sequence for and .

Note that if is a self-adjoint operator, then the two type of essential spectra agree by Weyl’s criterion, . Furthermore, in this case whenever .

### 2.4 Mellin transforms

For , let be its Mellin transform,

 Mg(ζ)=^g(ζ)=∫∞0tζg(t)dtt.

The -hypothesis on implies that is well-defined and bounded at least for . We will denote Mellin convolution by ,

 u⋆v(t)=∫∞0u(t/t′)v(t′)dt′t′,t>0.

The Mellin transform is the Fourier transform of the multiplicative group of ; for sufficiently nice functions and and appropriate it holds that

 M(u⋆v)(ζ)=Mu(ζ)Mv(ζ).

Young’s inequality for the Mellin transform says that

 ∥u⋆v∥L2(dt/t)≤∥u∥L1(dt/t)∥v∥L2(dt/t). (18)

Another way to see this is by noting that is a unitary operator,

 Wv(ξ)=1√2πMv(iξ), (19)

with inverse

 W−1ψ(t)=1√2π∫∞−∞t−iξψ(ξ)dξ.

In particular, Plancherel’s formula takes the form

 12π∫∞−∞∣∣∣∫∞0tiξv(t)dtt∣∣∣2dξ=∫∞0|v(t)|2dtt. (20)

### 2.5 Singular integral estimates on Sobolev spaces

Suppose first that is a Lipschitz graph

 Γ={(x,y,φ(x,y)):(x,y)∈R2}.

The parametrization then induces tangential derivatives on on . The (inhomogeneous) Sobolev space consists of those functions such that

 ∥f∥2H1(Γ)=∥f∥L2(Γ)+∥∂xf∥L2(Γ)+∥∂yf∥L2(Γ)<∞.

This also allows us to define in the case that is a bounded Lipschitz surface, via its Lipschitz manifold structure. In this setting, we will make use of the fact that is characterized by single layer potentials.

###### Lemma 2.6 ((51), Theorem 3.3).

Let be a bounded and simply connected Lipschitz surface. Then

 SΓ:L2(Γ)→H1(Γ)

is a continuous isomorphism.

For we define the Sobolev-Besov space via the Gagliardo-Slobodeckij norm

 ∥u∥2Hs(Γ)=∥u∥2L2(Γα)+∫Γ∫Γ|u(r)−u(r′)|2|r−r′|2+2sdσ(r)dσ(r′). (21)

The spaces , , coincide with the real interpolation scale between and , see for instance (50). For we define the space of distributions as the dual space of with respect to the scalar product of . Recall that coincides with the energy space , with equivalent norms.

Our goal in Section 4 is to view the operator , where has a single axially symmetric conical point, as a perturbation of , where is a straight cone. In doing so we will encounter many integral operators with weakly singular kernels. It is well known that such kernels generate compact operators, see for example (8) and (49). However, we have been unable to locate a precise statement which covers all of our cases. We therefore sketch a proof of a statement which is far from sharp, but sufficient for our purposes.

###### Lemma 2.7.

Let be a bounded and simply connected Lipschitz surface, and let be a kernel on satisfying

 |H(r,r′)|≲1|r−r′|, (22)
 |H(r,r′)−H(r∗,r′)|≲|r−r∗||r−r′|2,|r−r∗|<12|r−r′|, (23)

and

 |H(r′,r)−H(r′,r∗)|≲|r−r∗||r−r′|2,|r−r∗|<12|r−r′|. (24)

Then the integral operator

 Hf(r)=∫ΓH(r,r′)f(r′)dσ(r′)

defines compact operators , , and .

###### Proof.

For it is easy to show that the operator

 Gβf(r)=∫Γ1|r−r′|βf(r′)dσ(r′)

is bounded on , for instance by interpolating between the spaces and , on which the boundedness property is evident. Next, inequalities (22) and (23) imply that

 |H(r,r′)−H(r∗,r′)|≲|r−r∗|3/4|r−r′|7/4+|r−r∗|3/4|r∗−r′|7/4,r,r∗,r′∈Γ.

Hence, for

 |Hf(r)−Hf(r∗)|≲|r−r∗|3/4(G7/4|f|(r)+G7/4|f|(r∗)).

From this estimate we obtain that

 ∥Hf∥H1/2(Γ)≲2∫Γ[G7/4|f|(r)]2∫Γ1|r−r∗|3/2dσ(r∗)dσ(r)≲∫Γ[G7/4|f|(r)]2dσ(r)≲∥f∥L2(Γ).

Hence is bounded. In particular is compact, since is compactly contained in . By (24) the same argument yields that the -adjoint also maps into . Equivalently, by duality, maps into boundedly. Since is compactly contained in it follows that is compact. By duality, this is equivalent to saying that is compact. Since the statement of the lemma is symmetric with respect to and , it follows that also is compact. ∎

###### Remark 2.8.

If is a -surface, then satisfies the hypotheses of the lemma. Hence is a compact operator in this case (as is well known). Another example we have in mind is given by the kernel

 Hj(r,r′)=rj−r′


text:
### Theory:

Basic terminologies:
• $$m × n$$ Representation of rectangles.
• "$$m$$" is the number of rows in a rectangle.
• "$$n$$" is the number of columns in a rectangle.
Example:
$$3 × 7$$ rectangle:

Important!
• It is impossible to fill up the rectangle by using all the $$5$$-Tetromino types once, because of using the $$T$$-Tetromino which has its center square covered by the other three squares (in three sides).
• It is possible to fill up the rectangle when each of the $$5$$-Tetromino is used twice.


text:
# What is the depth of comparator circuit required in Gale Shapely and STCONN?

Stable matching problem and $$STCONN$$ can be solved using comparator circuits (refer https://arxiv.org/abs/1208.2721).

1. What is the depth of the $$CC$$ circuit necessary for stable matching? Is it in $$CC^1$$ or in other words $$O(\log (n))$$ depth comparator circuit?

2. What is the depth of the $$CC$$ circuit necessary for $$NL$$? Is it in $$CC^1$$ or in other words $$O(\log (n))$$ depth comparator circuit?

Notice $$STCONN$$ is complete for $$NL$$ and is in $$AC^1\subseteq NC^2$$ which perfect matching (not stable) is in $$NC^2$$ and is hard for $$NL$$.

If 1. and 2. and not in $$CC^1$$ then what is the minimum $$CC^i$$ upper bound?

• Have you tried looking at the proofs in the paper? They give some upper bounds. You shouldn't expect any lower bounds, since non-trivial lower bounds are difficult. State-of-the-art lower bounds can be found in Gál and Robere, Lower Bounds for (Non-monotone) Comparator Circuits – Yuval Filmus Jun 9 at 8:29
• I tried looking at the paper. I have no idea of what the upper bounds are. I am not seeking lower bounds. – Bread Winner Jun 9 at 9:18
• When I think of $NC$ I think of $NC^k$ at a particular $k$. The $NL$ reduction to $CC$ illustrated on pp. $35$ in arxiv.org/abs/1208.2721 for $n=5$ has at least $5$ layers and so depth for general $n$ is not appearing to be $polylogarithmic$. I similarly do not get insight for perfect matching. – Bread Winner Jun 9 at 9:28
• All the information you need is in the paper. It might require some work extracting it. – Yuval Filmus Jun 9 at 9:37
• The section on $NL\subseteq CC$ on pp. $35$ is straightforward but I still do not really see depth to be $o(n)$. – Bread Winner Jun 9 at 9:45


text:
# Math Help - Implicit Differentation

1. ## Implicit Differentation

$e^y+24-e^-1 = 5x^2+4y^2$

I can't figure this out....

I believe this stage is correct $e^y+e^-1=10x+8y*y'-y'$

But Need help on further solving it.

Help appreciate.

2. Actually the first step should be $y'e^y = 10x + 8yy'$

3. Where did you put the $e^-1$?

4. $e^{-1}$ is a constant and its derivative is 0.

5. I thought it was -e^-1.

Anyway, in case of constant I get $(10x)/(-1*e^y-8y)$. Where could I have gone wrong there?

6. I get $y' = \frac{10x}{e^y - 8y}$

7. The reason i have negative 1 is because there are two y', and when i place it ont he other side (-y') I factored and assumed 1. Why is this not correct?

Thanks however.


text:
1
JEE Main 2019 (Online) 10th April Evening Slot
+4
-1
A simple pendulum of length L is placed between the plates of a parallel plate capacitor having electric field E, as shown in figure. Its bob has mass m and charge q. The time period of the pendulum is given by :
A
$$2\pi \sqrt {{L \over {\sqrt {{g^2} - {{{q^2}{E^2}} \over {{m^2}}}} }}}$$
B
$$2\pi \sqrt {{L \over {\left( {g + {{qE} \over m}} \right)}}}$$
C
$$2\pi \sqrt {{L \over {\sqrt {{g^2} + {{{q^2}{E^2}} \over {{m^2}}}} }}}$$
D
$$2\pi \sqrt {{L \over {\left( {g - {{qE} \over m}} \right)}}}$$
2
JEE Main 2019 (Online) 10th April Morning Slot
+4
-1
Figure shows charge (q) versus voltage (V) graph for series and parallel combination of two given capacitors. The capacitances are :
A
40 $$\mu$$F and 10 $$\mu$$F
B
60 $$\mu$$F and 40 $$\mu$$F
C
20 $$\mu$$F and 30 $$\mu$$F
D
50 $$\mu$$F and 30 $$\mu$$F
3
JEE Main 2019 (Online) 9th April Evening Slot
+4
-1
The parallel combination of two air filled parallel plate capacitors of capacitance C and nC is connected to a battery of voltage, V. When the capacitors are fully charged, the battery is removed and after that a dielectric material of dielectric constant K is placed between the two plates of the first capacitor. The new potential difference of the combined system is :-
A
V
B
$${V \over {K + n}}$$
C
$${{(n+1)V} \over {K + n}}$$
D
$${{nV} \over {K + n}}$$
4
JEE Main 2019 (Online) 9th April Morning Slot
+4
-1
Determine the charge on the capacitor in the following circuit :
A
200μC
B
2μC
C
10μC
D
60μC
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text:
May you can help me undertanding the following conclusion: Suppose we have an agent who has preferences over contingent claims, represented by a concave function $U$. This simply means that $\mathbb{E}U(X)\le\mathbb{E}U(Y)$ where $X,Y$ represent two claims. Now suppose that if two contingent claims have the same mean and the same variance they are indifferent for the agent.
If we consider distributions concentrated on three points than the function $U$ must be quadratic.