Search

For two Kolmogorov spaces and a split map 𝒦 between the two, the twisted portion of the di-morphism is regular up to a certain convergent point 𝜃. Topologically, this is an accumulation point. Abstractly, this is the co-fibration of a spectral compass. We will discuss here the Goodwillie calculus and forcing notions for motivic Kolmogorov spaces.

For two Kolmogorov spaces and a split map 𝒦 between the two, the twisted portion of the di-morphism is regular up to a certain convergent point 𝜃. Topologically, this is an accumulation point. Abstractly, this is the co-fibration of a spectral compass. We will discuss here the Goodwillie calculus and forcing notions for motivic Kolmogorov spaces.

Let X be a smooth projective variety with a simple normal crossing divisor D := D-1 + D-2 + ... + D-n, where D-i subset of X are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks X-D,X-(r) over right arrow by constructing an I-function lying in a slice of Givental's Lagrangian cone for Gromov-Witten theory of multi-root stacks. We provide three applications: (1) We show that some genus zero invariants of X-D,X-(r) over right arrow stabilize for sufficiently large (r) over right arrow. (2) We state a generalized local-log-orbifold principle conjecture and prove a version of it. (3) We show that regularized quantum periods of Fano varieties coincide with classical periods of the mirror Landau-Ginzburg potentials using orbifold invariants of X-D,X-(r) over right arrow., Selecta Mathematica. N.S., 29 (1), ISSN:1420-9020, ISSN:1022-1824

This paper is devoted to systematically extend $f$-mirror symmetry between families of hypersurfaces in complete toric varieties, as introduced in \cite{R-fTV}, to families of complete intersections subvarieties. Namely, $f$-mirror symmetry is induced by framed duality of framed toric varieties extending Batyrev-Borisov polar duality between Fano toric varieties. Framed duality has been defined and essentially well described for families of hypersurfaces in toric varieties in the previous \cite{R-fTV}. Here it is developed for families of complete intersections, allowing us to strengthening some previous results on hypersurfaces. In particular, the class of projective complete intersections and their mirror partners are studied in detail. Moreover, a (generalized) Landau-Ginzburg/Complete-Intersection correspondence is discussed, extending to the complete intersection setup the LG/CY correspondence firstly studied Chiodo-Ruan and Krawitz., Comment: v4: Abstract and Introduction modified and some other minor modifications, according with the Referee's report. Final version to be published in the Journal of Geometry and Physics; 57 pages

PurposeThe purpose is to develop search and detection strategies that maximize the probability of detection of mine-like objects.Design/methodology/approachThe author have developed a methodology that incorporates variational calculus, number theory and algebra to derive a globally optimal strategy that maximizes the expected probability of detection.FindingsThe author found a set of look angles that globally maximize the probability of detection for a general class of mirror symmetric targets.Research limitations/implicationsThe optimal strategies only maximize the probability of detection and not the probability of identification.Practical implicationsIn the context of a search and detection operation, there is only a limited time to find the target before life is lost; hence, improving the chance of detection will in real terms be translated into the difference between success or failure, life or death. This rich field of study can be applied to mine countermeasure operations to make sure that the areas of operations are free of mines so that naval operations can be conducted safely.Originality/valueThere are two novel elements in this paper. First, the author determine the set of globally optimal look angles that maximize the probability of detection. Second, the author introduce the phenomenon of concordance between sensor images.

We prove the Landau-Ginzburg mirror symmetry conjecture between invertible quasi-homogeneous polynomial singularities at all genera. That is, we show that the FJRW theory (LG A-model) of such a polynomial is equivalent to the Saito-Givental theory (LG B-model) of the mirror polynomial.

Symmetry perception in dot patterns is tolerant to temporal delays of up to 60 ms within and between element pairs. However, it is not known how factors effecting symmetry discrimination in static patterns might affect temporal integration in dynamic patterns. One such feature is luminance polarity. Using dynamic stimuli with increasing temporal delay (SOA) between the onset of the first and second element in a symmetric pair, we investigated how four different luminance-polarity conditions affected the temporal integration of symmetric patterns. All four luminance polarity conditions showed similar upper temporal limits of approximately 60 ms. However psychophysical performance over all delay durations showed significantly higher symmetry thresholds for unmatched-polarity patterns at short delays, but also significantly less sensitivity to increasing temporal delay relative to matched-polarity patterns. These varying temporal windows are consistent with the involvement of a fast, sensitive first-order mechanism for matched-polarity patterns, and a slower, more robust second-order mechanism for unmatched-polarity patterns. Temporal integration windows for unmatched-polarity patterns were not consistent with performance expected from attentional mechanisms alone, and instead supports the involvement of second-order mechanisms that combines information from ON and OFF channels.

Topological superconductors are an essential component for topologically protected quantum computation and information processing. Although signatures of topological superconductivity have been reported in heterostructures, material realizations of intrinsic topological superconductors are rather rare. Here we use scanning tunnelling spectroscopy to study the transition metal dichalcogenide 4Hb-TaS2 that interleaves superconducting 1H-TaS2 layers with strongly correlated 1T-TaS2 layers, and find spectroscopic evidence for the existence of topological surface superconductivity. These include edge modes running along the 1H-layer terminations as well as under the 1T-layer terminations, where they separate between superconducting regions of distinct topological nature. We also observe signatures of zero-bias states in vortex cores. All the boundary modes exhibit crystallographic anisotropy, which—together with a finite in-gap density of states throughout the 1H layers—allude to the presence of a topological nodal-point superconducting state. Our theoretical modelling attributes this phenomenology to an inter-orbital pairing channel that necessitates the combination of surface mirror symmetry breaking and strong interactions. It, thus, suggests a topological superconducting state realized in a natural compound. Scanning tunnelling microscopy provides evidence for topological superconductivity in a naturally occurring transition metal dichalcogenide material.

Mirror symmetry breaking in systems composed of achiral molecules is of importance for the design of functional materials for technological applications as well as for the understanding of the mechanisms of spontaneous emergence of chirality. Herein, we report the design and molecular self‐assembly of two series of rod‐like achiral polycatenar molecules derived from a π‐conjugated 5,5’‐diphenyl‐2,2’‐bithiophene core with a fork‐like triple alkoxylated end and a variable single alkylthio chain at the other end. In both series of liquid crystalline materials, differing in the chain length at the trialkoxylated end, helical self‐assembly of the π‐conjugated rods in networks occurs, leading to wide temperature ranges (>200 K) of bicontinuous cubic network phases, in some cases being stable even around ambient temperatures. The achiral bicontinuous cubic Ia 3‾ d phase (gyroid) is replaced upon alkylthio chain elongation by a spontaneous mirror symmetry broken bicontinuous cubic phase (I23) and a chiral isotropic liquid phase (Iso1 [*]). Further chain elongation results in removing the I23 phase and the re‐appearance of the Ia 3‾ d phase with different pitch lengths. In the second series an additional tetragonal phase separates the two cubic phase types., Spontaneous mirror‐symmetry breaking. Achiral multichain π‐conjugated rod‐like molecules form a series of chiral and achiral liquid crystalline networks as well as mirror symmetry broken liquids. The effect of thioalkyl chains on helical self‐assembly and network formation is discussed.

There are many reports in the literature of bound states in the continuum (BICs) in systems with up–down mirror symmetry. Semiconductor-based technology requires bulk semiconductor substrates, which impose symmetry breaking in the vertical direction. In this paper, we explore the possibility of realizing BICs in a high refractive index subwavelength one-dimensional grating placed on a substrate with a refractive index that varies from 1 to almost the refractive index of the grating, while the refractive index above the grating is 1. We demonstrate that in gratings with broken up–down mirror symmetry not only symmetry-protected BICs can arise, but also Friedrich–Wintgen (FW) and interference-based (IB) BICs with diverging quality factors. The limit of the refractive index difference between the grating and the substrate supporting the BIC was found to be as little as 0.03. We also present a study of configurations composed of a finite number of grating stripes, with refractive indices corresponding to GaAs in the grating and Al-rich AlGaAs in the substrate. We demonstrate that such an all-semiconductor configuration enables Q-factors above 104 when composed of fewer than 20 periods and nearly exponential Q-factor growth with increasing numbers of grating periods. The results of this study pave the way for a new class of micro- and nano-optical cavities realised in standard all-semiconductor technology and relying on the high quality factor induced by BIC.

For all integers k > 0 k>0 , we prove that the hypergeometric function \[ I ^ k ( α ) = ∑ j = 0 ∞ ( ( 8 k + 4 ) j ) ! j ! ( 2 j ) ! ( ( 2 k + 1 ) j ) ! 2 ( ( 4 k + 1 ) j ) ! α j \widehat {I}_k(\alpha )=\sum _{j=0}^\infty \frac {\bigl ((8k+4)j\bigr )!j!}{(2j)!\bigl ((2k+1)j\bigr )!^2 \bigl ((4k+1)j\bigr )!} \ \alpha ^j \] is a period of a pencil of curves of genus 3 k + 1 3k+1 . We prove that the function I ^ k \widehat {I}_k is a generating function of Gromov–Witten invariants of the family of anticanonical del Pezzo hypersurfaces X = X 8 k + 4 ⊂ P ( 2 , 2 k + 1 , 2 k + 1 , 4 k + 1 ) X=X_{8k+4} \subset \mathbb {P}(2,2k+1,2k+1,4k+1) . Thus, the pencil is a Landau–Ginzburg mirror of the family. The surfaces X X were first constructed by Johnson and Kollár. The feature of these surfaces that makes our mirror construction especially interesting is that | − K X | = | O X ( 1 ) | = ∅ |-K_X|=|\mathcal {O}_X (1)|=\varnothing . This means that there is no way to form a Calabi–Yau pair ( X , D ) (X,D) out of X X and hence there is no known mirror construction for X X other than the one given here. We also discuss the connection between our construction and work of Beukers, Cohen and Mellit on hypergeometric functions.

Cataloged from PDF version of article. Thesis (Ph.D.): Bilkent University, Department of Electrical and Electronics Engineering, İhsan Doğramacı Bilkent University, 2022. Includes bibliographical references (leaves 68-85). Magnetic particle imaging (MPI) is a rapidly developing medical imaging modality that exploits the non-linear response of magnetic nanoparticles (MNPs). Color MPI widens the functionality of MPI, empowering it with the capability to distinguish different MNPs and/or MNP environments. The system function approach for color MPI relies on extensive calibrations that capture the differences in the harmonic responses of the MNPs. An alternative calibration-free x-space-based method called TAURUS (TAU estimation via Recovery of Underlying mirror Symmetry) estimates a map of the relaxation time constant, τ, by recovering the underlying mirror symmetry in the MPI signal. However, TAURUS requires a back and forth scanning of a given region, restricting its usage to slow trajectories with constant or piecewise constant focus fields (FFs). In this thesis, a novel technique is proposed to increase the performance of TAURUS and enable τ map estimation for rapid and multi-dimensional trajectories. The proposed technique is based on correcting the distortions on mirror symmetry induced by time-varying FFs. Simulations and experiments in an in-house MPI scanner demonstrates that the proposed method successfully estimates high-fidelity τ maps for rapid trajectories that provide orders of magnitude reduction in scanning time (over 300 fold for simulations and over 8 fold for experiments) while preserving the calibration-free property of TAURUS. Additionally, the proposed method can successfully map the effective relaxation time constants in a relatively wide field-of-view at frame rates exceeding 5 frames-per-second. This thesis presents the first simulation results demonstrating that the proposed method is capable of generating high fidelity and high frame-rate color MPI images in real time. by Musa Tunç Arslan Ph.D.

For any triple of positive integers $A^{\prime} = (a_1^{\prime},a_2^{\prime},a_3^{\prime})$ and $c \in{{\mathbb{C}}}^*$, cusp polynomial ${ f_{A^\prime }} = x_1^{a_1^{\prime}}+x_2^{a_2^{\prime}}+x_3^{a_3^{\prime}}-c^{-1}x_1x_2x_3$ is known to be mirror to Geigle–Lenzing orbifold projective line ${{\mathbb{P}}}^1_{a_1^{\prime},a_2^{\prime},a_3^{\prime}}$. More precisely, with a suitable choice of a primitive form, the Frobenius manifold of a cusp polynomial ${ f_{A^\prime }}$ turns out to be isomorphic to the Frobenius manifold of the Gromov–Witten theory of ${{\mathbb{P}}}^1_{a_1^{\prime},a_2^{\prime},a_3^{\prime}}$. In this paper we extend this mirror phenomenon to the equivariant case. Namely, for any $G$—a symmetry group of a cusp polynomial ${ f_{A^\prime }}$, we introduce the Frobenius manifold of a pair$({ f_{A^\prime }},G)$ and show that it is isomorphic to the Frobenius manifold of the Gromov–Witten theory of Geigle–Lenzing weighted projective line ${{\mathbb{P}}}^1_{A,\Lambda }$, indexed by another set $A$ and $\Lambda $, distinct points on ${{\mathbb{C}}}\setminus \{0,1\}$. For some special values of $A^{\prime}$ with the special choice of $G$ it happens that ${{\mathbb{P}}}^1_{A^{\prime}} \cong{{\mathbb{P}}}^1_{A,\Lambda }$. Combining our mirror symmetry isomorphism for the pair $(A,\Lambda )$, together with the “usual” one for $A^{\prime}$, we get certain identities of the coefficients of the Frobenius potentials. We show that these identities are equivalent to the identities between the Jacobi theta constants and Dedekind eta–function.

Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of "theta functions" on the coordinate rings of various log Calabi-Yau spaces, including cluster varieties. We prove that the theta bases for cluster varieties are determined by certain descendant log Gromov-Witten invariants of the symplectic leaves of the mirror/Langlands dual cluster variety, as predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We further show that these Gromov-Witten counts are often given by naive counts of rational curves satisfying certain geometric conditions. As a key new technical tool, we introduce the notion of "contractible" tropical curves when showing that the relevant log curves are torically transverse., Comment: 36 pages, 2 figures; published version

This is the first part of our project toward proving the Bershadsky–Cecotti–Ooguri–Vafa Feynman graph sum formula of all genera Gromov–Witten invariants of quintic Calabi–Yau threefolds. We introduce the notion of N–mixed-spin-P fields, construct their moduli spaces, their virtual cycles and their virtual localization formulas, and obtain a vanishing result associated with irregular graphs.

Among the most actively studied issues in the cuprates are the natures of the pseudogap and strange metal states and their relationship to superconductivity1. There is general agreement that the low-energy physics of the Mott-insulating parent state is well captured by a two-dimensional spin S = 1/2 antiferromagnetic Heisenberg model2. However, recent observations of a large thermal Hall conductivity in several parent cuprates appear to defy this simple model and suggest proximity to a magneto-chiral state that breaks all mirror planes that are perpendicular to the CuO2 layers3–6. Here we use optical second harmonic generation to directly resolve the point group symmetries of the model parent cuprate Sr2CuO2Cl2. We report evidence of an order parameter that breaks all perpendicular mirror planes and is consistent with a magneto-chiral state in zero magnetic field. Although this order is clearly coupled to the antiferromagnetism, we are unable to realize its time-reversed partner by thermal cycling through the antiferromagnetic transition temperature or by sampling different spatial locations. This suggests that the order onsets above the Neel temperature and may be relevant to the mechanism of pseudogap formation. Several cuprate superconductors were recently shown to have chiral phonons. Here second harmonic generation measurements show that antiferromagnetism breaks all mirror symmetries in a related compound, consistent with a chiral state.

We evaluate half-indices of $\mathcal{N}=(2,2)$ half-BPS boundary conditions in 3d $\mathcal{N}=4$ supersymmetric Abelian gauge theories. We confirm that the Neumann boundary condition is dual to the generic Dirichlet boundary condition for its mirror theory as the half-indices perfectly match with each other. We find that a naive mirror symmetry between the exceptional Dirichlet boundary conditions defining the Verma modules of the quantum Coulomb and Higgs branch algebras does not always hold. The triangular matrix obtained from the elliptic stable envelope describes the precise mirror transformation of a collection of half-indices for the exceptional Dirichlet boundary conditions., Comment: 51 pages, v3: published version in JHEP

We consider a waveguide composed of two not necessity equal semi-infinite strips (trunks) and a rectangle (resonator) connected by narrow openings in the shared walls. As their diameter vanishes, we construct asymptotics for the scattering coefficients, justifying them by the technique of weighted spaces with detached asymptotics. We establish a criterion for the possibility of observing, at a given frequency, almost complete transmission of waves through both perforated cross-walls. This effect is revealed due to a fine-tuning of the resonator height and the criterion involves an equation relating some geometrical characteristics of the waveguide to the wave numbers in the trunks, while any mirror symmetry turns the criterion trivial.

We consider a pair of quiver varieties $(X;X^{\prime})$ related by 3D mirror symmetry, where $X =T^*{Gr}(k,n)$ is the cotangent bundle of the Grassmannian of $k$-planes of $n$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on $X \times X^{\prime} $ (the mother function) whose restrictions to $X$ and $X^{\prime} $ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for $X$ and $X^{\prime}$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.

We review some classical and modern aspects of hypergeometric differential equations, including A-hypergeometric systems of Gel$$'$$ ′ fand, Graev, Kapranov and Zelevinsky. Some recent advances in this theory, such as Euler–Koszul homology, rank jump phenomena, irregularity questions and Hodge theoretic aspects are discussed with more details. We also give some applications of the theory of hypergeometric systems to toric mirror symmetry.

The stacking fault structure (SFT) is the key to understanding the symmetry breaking of fcc nanocrystals and the origin of two-dimensional (2D) anisotropic growth of nanoplates. After resolving the SFT in Ag nanoplates under aberration-corrected transmission electron microscope (TEM) observations, it is found that there are three basic stacking faults, namely, twinned stacking fault (SF-t), a layer missed stacking fault (SF-m) and a layer inserted stacking fault (SF-i). The SFT is composed of one or a combination of two or all of the three kinds of stacking faults with a total number varying from 4 to 9. It has been demonstrated that the SFT could generate concave faces, step faces and (100) faces in the lateral directions, which provides sites for adding-atoms with a higher coordination number than on the top and bottom flat (111) faces, and results in the anisotropic growth along the 2D direction. Additionally, Ag nanoplates fall into either center symmetry or mirror symmetry when the corresponding number is even or odd. The center symmetry and mirror symmetry with different side face arrangements in turn manipulate the shape evolution to cubes and bipyramids, respectively. Our study provides a comprehensive understanding of the formation and growth of 2D metal nanomaterials.

In this paper, we point out an interesting asymmetry in the rules of fundamental mathematics between positive and negative numbers. Further, we show an alternative numerical system identical to today’s system, but where positive numbers dominate over negative numbers. This is like a mirror symmetry of the existing number system. The asymmetry in both systems leads to imaginary and complex numbers. We also suggest an alternative number system with perfectly symmetrical rules—that is, where there is no dominance of negative numbers over positive numbers or vice versa, and where imaginary and complex numbers are no longer needed. This number system seems to be superior to other numerical systems, as it brings simplicity and logic back to areas that complex rules have dominated for much of the history of mathematics. Finally, we also briefly discuss how the Riemann hypothesis may be linked to the asymmetry in the current number system. The foundation rules of a number system can, in general, not be proven incorrect or correct inside the number system itself. However, the ultimate goal of a number system is, in our view, to describe nature accurately. The optimal number system should therefore be developed with feedback from nature. If nature, at a very fundamental level, is ruled by symmetry, then a symmetric number system should make it easier to understand nature than an asymmetric number system would. We hypothesize that a symmetric number system may thus be better suited to describing nature. Further, such a number system should eliminate imaginary numbers in space-time and quantum mechanics; for example, two areas of physics that are clouded in mystery to this day.

The crystallographic analysis of the structure of BaV6O11, a representative of the АV6O11 (А = Na, K, Sr, Pb, Ba) family, allows us to establish its similarity with the structures of SrAl12O19 and BaFe18O27 polytypes with conjugated cation and anion (together with a large cation) sublattices that have common nodal planes in the mirror symmetry planes perpendicular to the hexagonal axis. The relative stability of the structures is estimated from their composition and symmetry.

Ideal topological materials are those stable materials with less nontrivial band crossing near the Fermi surface and a long Fermi arc. By means of first-principles calculations, here we present that the 3D monochalcogenide molybdenum telluride (Pm-MoTe) without an inversion center shows a type-II Weyl semimetal (WSM) phase which cannot checked by symmetry index method. A total of eight Weyl points (WPs) are found in different quadrants of the Brillouin zone (BZ) of Pm-MoTe, which guarantee a long Fermi arc. The WSM phase is robust against the spin–orbit coupling (SOC) effect because of mirror symmetry and time reversal symmetry. It is also found that a topological phase transition can be tuned by strain. For different types of strain, the number of WPs can be effectively modulated to a minimum number, and their energies could be closer to Fermi level. These findings propose a promising material candidate that partly satisfies the ideal WSM criteria and extends the potential applications of the tunable topological phase.

These notes were born out of a five-hour lecture series for graduate students at the May 2018 Snowbird workshop Crossing the Walls in Enumerative Geometry. After a short primer on equivariant cohomology and localization, we provide proofs of the genus-zero mirror theorems for the quintic threefold, first in Fan-Jarvis-Ruan-Witten theory and then in Gromov-Witten theory. We make no claim to originality, except in exposition, where special emphasis is placed on peeling away the standard technical machinery and viewing the mirror theorems as closed-formula manifestations of elementary localization recursions., Comment: 24 pages, comments welcome

Majorana fermions in two-dimensional systems satisfy non-Abelian statistics. They are possible to exist in topological superconductors as quasi particles, which is of great significance for topological quantum computing. In this paper, we study a new promising system of superconducting topological surface state topological insulator thin films. We also study the phase diagrams of the model by plotting the Majorana edge states and the density of states in different regions of the phase diagram. Due to the mirror symmetry of the topological surface states, the Hamiltonian can be block diagonalized into two spin-triplet p-wave superconductors, which are also confirmed by the phase diagrams. The chiral Majorana edge modes may provide a new route for realizing topological quantum computation.

We study mirror symmetry of families of elliptic K3 surfaces with elliptic fibers of type $E_6,~E_7$ and $E_8$. We consider a moduli space $\mathsf{T}$ of the mirror K3 surfaces enhanced with the choice of differential forms. We show that coordinates on $\mathsf{T}$ are given by the ring of quasi modular forms in two variables, with modular groups adapted to the fiber type. We furthermore introduce an algebraic group $\mathsf{G}$ which acts on $\mathsf{T}$ from the right and construct its Lie algebra $\mathrm{Lie}(\mathsf{G})$. We prove that the extended Lie algebra generated by $\mathrm{Lie}(\mathsf{G})$ together with modular vector fields on $\mathsf{T}$ is isomorphic to $\mathrm{sl}_2(\mathbb{C})\oplus\mathrm{sl}_2(\mathbb{C})$., 19 pages

Inspired by mirror symmetry, we investigate some differential geometric aspects of the space of Bridgeland stability conditions on a Calabi-Yau triangulated category. The aim is to develop theory of Weil-Petersson geometry on the stringy Kahler moduli space. A few basic examples are studied. In particular, we identify our Weil-Petersson metric with the Bergman metric on a Siegel modular variety in the case of the self-product of an elliptic curve.

The diffraction features of Si(111) 7 × 7 are analyzed and related to various structural models of the Si(111) 7 × 7 surface as one part of a multivariate analysis of this system. The limitations in early sample preparation and measurements produce some uncertainty in previously proposed structures. More recent data is considered here. In addition, models used early on to evaluate the structure of 7 × 7 have been over simplified, idealized models. More complex models are considered within the projection rod method as used for surface crystallography. The origin of numerous diffraction features can be determined via their Fourier components for a wide range of 2-D layers, which provides new insight into the structure as well as the limitations of prior projection analyses. Structures which produce the key elements of the 7 × 7 diffraction are presented and various distortions are considered consistent with other experimental results. In general it is found that the presence of a strong set of 3/7th order beams and near extinction of neighbouring fractional order beams are features which are found experimentally and distinguish an important class of structures. This class has a particular type of 3-fold mirror symmetry, which is not apparent in the widely accepted dimer-adatom-stacking fault, DAS, model. Higher order diffraction features, of which many are weak, are also considered and provide important new structural information. Several new polymorphs of the 7 × 7 are identified which may also satisfy the diffraction derived features and possess some degree of pi-bonding so as to enable magnetic surface states not possible in a pure covalently bonded system such as DAS. The Patterson map of the 7 × 7 surface provide insight into the lost phase information encoded in diffraction and reveal why the DAS structure was experimentally favored. An unusual non-primitive 7 × 7 unit cell is also derived from the Patterson map that possesses unusual symmetry properties, a non-standard surface Brillouin zone with potentially unusual electronic properties.

To a one-parameter family of Calabi-Yau threefolds, we can associate the extended period map by the log Hodge theory of Kato and Usui. In the present paper, we study the image of a maximally unipotent monodromy point under the extended period map. As an application, we prove the generic Torelli theorem for a large class of one-parameter families of Calabi-Yau threefolds.

The analysis of the geometric and algebraic properties of mirror mappings allowed the latter to be used as the operator algebra of a noncommutative geometry. The coordinates of the noncommutative geometry are auto- or cross-correlation coordinates in the mirror-mapped spaces. A particular case of the six-dimensional Kahler manifold which is mapped on the noncommutative geometry with the vector Clifford algebra Cl4 has been considered. This mapping corresponds to a tetraquark composed from two quark–anti-quark pairs with the charges ±2/3q taken from different generations., Аналiз геометричних i алгебраїчних властивостей дзеркальних вiдображень дозволив використовувати їх як операторну алгебру некомутативної геометрiї. Координатами некомутативної геометрiї є авто- або крос-кореляцiї координат дзеркально вiдображених просторiв. Розглянуто окремий випадок шестивимiрного келерова многовиду, який вiдображається на некомутативну геометрiю з векторної алгеброю Клiффорда Cl4 . Цей випадок вiдображення вiдповiдає тетракварку у складi двох пар кварк-антикварк з рiзних поколiнь i зарядами ±2/3q.

Two Kirchhoff plates are considered that are joined by means of two rows of rivets posed with the small period h and modeled by Sobolev point transmission conditions. A new and unexpected effect is observed, namely the almost complete clutch of the plates occurs even for an exponentially small (with respect to h) distance between the rows in the mirror symmetry case. The same effect appears in the case of rivets in chessboard order only for the power-law smallness of the distance. These results are obtained with the help of analysis of boundary layer phenomena. Conditions are found to provide the hinge joint of the plates with friction at the limit.

In this communication, a switchable active metasurface (SA-MS), with functions for absorption and polarization conversion, is proposed. The metasurface comprises rotated cross loops that are loaded with four sets of parallel-connected lumped resistors and p-i-n diodes on each arm. When the p-i-n diodes are in the OFF-state, the proposed structure performs broadband absorption functions from 3.7 to 10.7 GHz. When the p-i-n diodes are in the ON-state, those parallel lumped resistors are shorted and the structure transforms into a rectangular loop that converts the $x(y)$ - to a $y(x)$ -polarized one from 5.9 to 9.6 GHz. Furthermore, the proposed SA-MS and its mirror symmetry structure are used as the meta-atoms in the chessboard array. Such implementation can achieve backward radar cross section (RCS) reduction based on the phase cancellation principle in the conversion mode without any impact on the absorption function. Moreover, the scattering pattern of the checkerboard structure in the transition mode is more varied than that in the absorption mode. The two different modes provide flexibility for the RCS reduction and enable it to cope with complex electronic reconnaissance. Finally, a chessboard array and uniform array sample printed on the FR4 dielectric layer have been fabricated and measured. The experimental results are consistent with the simulated results.

We consider the phenomenon of the complete coincidence of key properties of Calabi–Yau manifolds realized as hypersurfaces in two different weighted projective spaces. More precisely, the first manifold in such a pair is realized as a hypersurface in a weighted projective space, and the second is realized as a hypersurface in an orbifold of another weighted projective space. The two manifolds in each pair have the same Hodge numbers and the same geometry on the complex structure moduli space and are also associated with the same $$N{=}2$$ gauged linear sigma model. We explain these coincidences using the correspondence between Calabi–Yau manifolds and the Batyrev reflexive polyhedra.

The recently conjectured knots-quivers correspondence (Kucharski et al. in Phys Rev D 96(12):121902, 2017. arXiv:1707.02991 , Adv Theor Math Phys 23(7):1849–1902, 2019. arXiv:1707.04017 ) relates gauge theoretic invariants of a knot K in the 3-sphere to the representation theory of a quiver $$Q_{K}$$ associated to the knot. In this paper we provide geometric and physical contexts for this conjecture within the framework of Ooguri-Vafa large N duality (Ooguri and Vafa in Nucl Phys B 577:419–438, 2000), that relates knot invariants to counts of holomorphic curves with boundary on $$L_{K}$$ , the conormal Lagrangian of the knot in the resolved conifold, and corresponding M-theory considerations. From the physics side, we show that the quiver encodes a 3d $${\mathcal {N}}=2$$ theory $$T[Q_{K}]$$ whose low energy dynamics arises on the worldvolume of an M5 brane wrapping the knot conormal and we match the (K-theoretic) vortex partition function of this theory with the motivic generating series of $$Q_{K}$$ . From the geometry side, we argue that the spectrum of (generalized) holomorphic curves on $$L_{K}$$ is generated by a finite set of basic disks. These disks correspond to the nodes of the quiver $$Q_{K}$$ and the linking of their boundaries to the quiver arrows. We extend this basic dictionary further and propose a detailed map between quiver data and topological and geometric properties of the basic disks that again leads to matching partition functions. We also study generalizations of A-polynomials associated to $$Q_{K}$$ and (doubly) refined version of LMOV invariants (Ooguri and Vafa 2000; Labastida and Marino in Commun Math Phys 217(2):423–449, 2001. arXiv:hep-th/0004196 ; Labastida et al. in JHEP 11:007, 2000. arXiv:hep-th/0010102 ; Aganagic and Vafa in Large N duality, mirror symmetry, and a Q-deformed A-polynomial for knots. arXiv:1204.4709 ; Fuji et al. in Nucl Phys B 867:506–546, 2013. arXiv:1205.1515 ).

We present experimental data on water repetitively brought in contact with cellophane. Although this hydrophilic polymer is insoluble in water, repetitively immersing it in water changes the liquid’s properties. We compare the physicochemical properties of the water left over after removing the cellophane to those of previously published data on water repetitively brought in contact with other in water insoluble polymers (cotton wool or Nafion). Some of the properties are similar. All the properties considerably differ from those of the Milli-Q® water used. On lyophilizing these waters, solid residues remain. The residues are soluble in water. The chemical nature of the residues differs from that of the perturbing polymers.

We investigate simple extensions of the Mirror Twin Higgs model in which the twin color gauge symmetry and the discrete $Z_2$ mirror symmetry are spontaneously broken. This is accomplished in a minimal way by introducing a single new colored triplet, sextet, or octet scalar field and its twin along with a suitable scalar potential. This spontaneous $Z_2$ breaking allows for a phenomenologically viable alignment of the electroweak vacuum, and leads to dramatic differences between the visible and mirror sectors with regard to the residual gauge symmetries at low energies, color confinement scales, and particle spectra. In particular, several of our models feature a remnant $SU(2)$ or $SO(3)$ twin color gauge symmetry with a very low confinement scale in comparison to $\Lambda_{\rm QCD}$. Furthermore, couplings between the colored scalar and matter provide a new dynamical source of twin fermion masses, and due to the mirror symmetry, these lead to a variety of correlated visible sector effects that can be probed through precision measurements and collider searches., Comment: 45 pages, 2 figures; Updated to published version

In this paper, we describe recent work towards the mirror $$\mathrm P=\mathrm W$$ conjecture, which relates the weight filtration on the cohomology of a log Calabi–Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual log Calabi–Yau manifold taken with respect to the affinization map. This conjecture extends the classical relationship between Hodge numbers of mirror dual compact Calabi–Yau manifolds, incorporating tools and ideas which appear in the fascinating and groundbreaking works of de Cataldo, Hausel, and Migliorini [1] and de Cataldo and Migliorini [2]. We give a broad overview of the motivation for this conjecture, recent results towards it, and describe how this result might arise from the SYZ formulation of mirror symmetry. This interpretation of the mirror $$\mathrm P=\mathrm W$$ conjecture provides a possible bridge between the mirror $$\mathrm P=\mathrm W$$ conjecture and the well-known $$\mathrm P=\mathrm W$$ conjecture in non-Abelian Hodge theory.

We compute the $g=1$, $n=1$ B-model Gromov–Witten invariant of an elliptic curve $E$ directly from the derived category $\mathsf{D}_{\mathsf{coh}}^{b}(E)$. More precisely, we carry out the computation of the categorical Gromov–Witten invariant defined by Costello using as target a cyclic $\mathscr{A}_{\infty }$ model of $\mathsf{D}_{\mathsf{coh}}^{b}(E)$ described by Polishchuk. This is the first non-trivial computation of a positive-genus categorical Gromov–Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov–Witten invariants of a symplectic 2-torus computed by Dijkgraaf.

Van der Waals heterostructures form a massive interdisciplinary research field, fueled by the rich material science opportunities presented by layer assembly of artificial solids with controlled composition, order and relative rotation of adjacent atomic planes. Here we use atomic resolution transmission electron microscopy and multiscale modeling to show that the lattice of MoS$_2$ and WS$_2$ bilayers twisted to a small angle, $\theta

By mirror symmetry, the quantum connection of a weighted projective line is closely related to the localized Fourier–Laplace transform of some Gauß–Manin system. Following an article of D’Agnolo, Hien, Morando, and Sabbah, we compute the Stokes matrices for the latter at $$\infty $$ ∞ for the cases $${\mathbb {P}}(1,3)$$ P ( 1 , 3 ) and $${\mathbb {P}}(2,2)$$ P ( 2 , 2 ) by purely topological methods. We compare them to the Gram matrix of the Euler–Poincaré pairing on $$D^b(\mathrm{Coh}({\mathbb {P}}(1,3)))$$ D b ( Coh ( P ( 1 , 3 ) ) ) and $$D^b(\mathrm{Coh}({\mathbb {P}}(2,2)))$$ D b ( Coh ( P ( 2 , 2 ) ) ) , respectively. This article is based on the doctoral thesis of the author.

Conservation laws are deeply related to any symmetry present in a physical system1,2. Analogously to electrons in atoms exhibiting spin symmetries3, it is possible to consider neutrons and protons in the atomic nucleus as projections of a single fermion with an isobaric spin (isospin) of t = 1/2 (ref. 4). Every nuclear state is thus characterized by a total isobaric spin T and a projection Tz—two quantities that are largely conserved in nuclear reactions and decays5,6. A mirror symmetry emerges from this isobaric-spin formalism: nuclei with exchanged numbers of neutrons and protons, known as mirror nuclei, should have an identical set of states7, including their ground state, labelled by their total angular momentum J and parity π. Here we report evidence of mirror-symmetry violation in bound nuclear ground states within the mirror partners strontium-73 and bromine-73. We find that a J π = 5/2− spin assignment is needed to explain the proton-emission pattern observed from the T = 3/2 isobaric-analogue state in rubidium-73, which is identical to the ground state of strontium-73. Therefore the ground state of strontium-73 must differ from its J π = 1/2− mirror bromine-73. This observation offers insights into charge-symmetry-breaking forces acting in atomic nuclei. Observations of the decay of 73Sr, when compared to its mirror nucleus 73Br, indicate that the spin assignment of their ground states differ, demonstrating mirror-symmetry violation.

Theoretical physics is taking an increasing part in the universe of mathematics. After calculus, vector and tensorial analysis, topological theories make their entry into quantum field theories. More precisely, in this domain, topological theories are the most relevant. A fundamental theorem of the Atiyah has important repercussions in several branches of quantum physics in the geometric approach. We can cite the work of Alain Connes on non-commutative geometry, but also all the developments due to Donaldson, E. Witten around Gauge theories, superstring and Mirror Symmetry. We present here an historical survey of some topological field theories, especially Mirror Symmetry to understand the interpenetration between quantum physics and topology.

A compact complex manifold is Kobayashi non-hyperbolic if there exists an entire curve on it. Using mirror symmetry we establish that there are (possibly singular) elliptic or rational curves on any Calabi-Yau manifold $X$, whose mirror dual $\check X$ exists and is not "Hodge degenerate", therefore proving that $X$ is Kobayashi non-hyperbolic. We are not aware of any higher dimensional simply connected Calabi-Yau manifolds that satisfy the "Hodge degenerate" condition., Comment: 8 pages, comments are welcome

The Remodeling Conjecture proposed by Bouchard-Klemm-Marino-Pasquetti [arXiv:0709.1453, arXiv:0807.0597] relates all genus open and closed Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-manifolds/3-orbifolds to the Eynard-Orantin invariants of the mirror curve of the toric Calabi-Yau 3-fold. In this paper, we present a proof of the Remodeling Conjecture for open-closed orbifold Gromov-Witten invariants of an arbitrary affine toric Calabi-Yau 3-orbifold relative to a framed Aganagic-Vafa Lagrangian brane. This can be viewed as an all genus open-closed mirror symmetry for affine toric Calabi-Yau 3-orbifolds.

Background Previously, a seven-cluster pattern claiming to be a universal one in bacterial genomes has been reported. Keeping in mind the most popular theory of chloroplast origin, we checked whether a similar pattern is observed in chloroplast genomes. Results Surprisingly, eight cluster structure has been found, for chloroplasts. The pattern observed for chloroplasts differs rather significantly, from bacterial one, and from that latter observed for cyanobacteria. The structure is provided by clustering of the fragments of equal length isolated within a genome so that each fragment is converted in triplet frequency dictionary with non-overlapping triplets with no gaps in frame tiling. The points in 63-dimensional space were clustered due to elastic map technique. The eight cluster found in chloroplasts comprises the fragments of a genome bearing tRNA genes and exhibiting excessively high GC-content, in comparison to the entire genome. Conclusion Chloroplasts exhibit very specific symmetry type in distribution of coding and non-coding fragments of a genome in the space of triplet frequencies: this is mirror symmetry. Cyanobacteria may have both mirror symmetry, and the rotational symmetry typical for other bacteria.

We give a short new computation of the quantum cohomology of an arbitrary smooth toric variety $X$, by showing directly that the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono defines an isomorphism onto a suitable Jacobian ring. The proof is based on the purely algebraic fact that a class of generalised Jacobian rings associated to $X$ are free as modules over the Novikov ring. In contrast to previous results of this kind, $X$ need not be compact. When $X$ is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure., This article has been accepted for publication in The Quarterly Journal of Mathematics Published by Oxford University Press. The accepted version is available on my webpage