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• 1.
Virginia Commonwealth University, USA.
Stable magnetic order and charge induced rotation of magnetizationin nano-clusters2014In: Applied Physics Letters, ISSN 0003-6951, E-ISSN 1077-3118, Vol. 105, no 152409Article in journal (Refereed)

Efficient control of magnetic anisotropy and the orientation of magnetization are of centralimportance for the application of nanoparticles in spintronics. Conventionally, magnetization iscontrolled directly by an external magnetic field or by an electric field via spin-orbit coupling.Here, we demonstrate a different approach to control magnetization in small clusters. We firstshow that the low magnetic anisotropy of a Co5 cluster can be substantially enhanced by attachingbenzene molecules due to the mixing between p states of C and the d states of Co sites. We thenshow that the direction of magnetization vector of Co5 sandwiched between two benzene moleculesrotates by 90 when an electron is added or removed from the system. An experimental set up torealize such effect is also suggested.

• 2.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Ab initio calculations of the magnetic properties of Mn impurities on GaAs (110) surfaces2012In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. B 85, p. Article ID: 155306-Article in journal (Refereed)

We present a computational study of individual and pairs of substitutional Mn impurities on the (110) surface of GaAs samples based on density functional theory. We focus on the anisotropy properties of these magnetic centers and their dependence on on-site correlations, spin-orbit interaction, and surface-induced symmetry-breaking effects. For a Mn impurity on the surface, the associated acceptor-hole wave function tends to be more localized around the Mn than for an impurity in bulk GaAs. The magnetic anisotropy energy for isolated Mn impurities is of the order of 1 meV, and can be related to the anisotropy of the orbital magnetic moment of the Mn acceptor hole. Typically Mn pairs have their spin magnetic moments parallel aligned, with an exchange energy that strongly depends on the pair orientation on the surface. The spin magnetic moment and exchange energies for these magnetic entities are not significantly modified by the spin-orbit interaction, but are more sensitive to on-site correlations. Correlations in general reduce the magnetic anisotropy for most of the ferromagnetic Mn pairs.

• 3.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. NORDITA, Stockholm. NORDITA, Stockholm. CNR, Italy. CNR, Italy. CNR, Italy. Novosibirsk State Univ, Russia. Novosibirsk State Univ, Russia. European Synchrotron Radiat Facil, France. European Synchrotron Radiat Facil, France. ALBA Synchrotron Light Source, Spain. ALBA Synchrotron Light Source, Spain. Univ Wurzburg, Germany. Univ Wurzburg, Germany. Univ Wurzburg, Germany. Univ Wurzburg, Germany. Univ Wurzburg, Germany. Univ Wurzburg, Germany. Univ Wurzburg, Germany. Univ Wurzburg, Germany. Univ Wurzburg, Germany.
Systematics of electronic and magnetic properties in the transition metal doped Sb2Te3 quantum anomalous Hall platform2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 97, no 15, article id 155429Article in journal (Refereed)

The quantum anomalous Hall effect (QAHE) has recently been reported to emerge in magnetically doped topological insulators. Although its general phenomenology is well established, the microscopic origin is far from being properly understood and controlled. Here, we report on a detailed and systematic investigation of transition metal (TM) doped Sb2Te3. By combining density functional theory calculations with complementary experimental techniques, i.e., scanning tunneling microscopy, resonant photoemission, and x-raymagnetic circular dichroism, we provide a complete spectroscopic characterization of both electronic and magnetic properties. Our results reveal that the TM dopants not only affect the magnetic state of the host material, but also significantly alter the electronic structure by generating impurity-derived energy bands. Our findings demonstrate the existence of a delicate interplay between electronic and magnetic properties in TM doped topological insulators. In particular, we find that the fate of the topological surface states critically depends on the specific character of the TM impurity: while V-and Fe-doped Sb2Te3 display resonant impurity states in the vicinity of the Dirac point, Cr and Mn impurities leave the energy gap unaffected. The single-ion magnetic anisotropy energy and easy axis, which control the magnetic gap opening and its stability, are also found to be strongly TM impurity dependent and can vary from in plane to out of plane depending on the impurity and its distance from the surface. Overall, our results provide general guidelines for the realization of a robust QAHE in TM doped Sb2Te3 in the ferromagnetic state.

• 4.
Virginia Commonwealth University, USA.
Virginia Commonwealth University, USA.
On the enhancement of magnetic anisotropy in cobalt clusters via non-magnetic doping2014In: Journal of Physics: Condensed Matter, ISSN 0953-8984, E-ISSN 1361-648X, Vol. 26, no 125303Article in journal (Refereed)

We show that the magnetic anisotropy energy (MAE) in cobalt clusters can be significantlyenhanced by doping them with group IV elements. Our firstprincipleselectronic structurecalculations show that Co4C2 and Co12C4 clusters have MAEs of 25 K and 61 K, respectively. The large MAE is due to controlled mixing between Co dandC pstatesand can be furthertuned by replacing C by Si. Larger assemblies of such primitive units are shown to be stablewith MAEs exceeding 100 K in units as small as 1.2 nm, in agreement with the recentobservation of large coercivity. These results may pave the way for the use of nanoclustersinhigh density magnetic memory devices for spintronics applications

• 5.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics. Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics. Naval Research Laboratory, USA.
First-principles study of spin-electric coupling in a {Cu3} single molecular magnet2010In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 82, no 15, p. article number-155446Article in journal (Refereed)

We report on a study of the electronic and magnetic properties of the triangular antiferromagnetic {Cu3} single-molecule magnet, based on spin-density-functional theory. Our calculations show that the low-energy magnetic properties are correctly described by an effective three-site spin s = 1/2 Heisenberg model, with an antiferromagnetic exchange coupling J approximate to 5 meV. The ground-state manifold of the model is composed of two degenerate spin S = 1/2 doublets of opposite chirality. Due to lack of inversion symmetry in the molecule these two states are coupled by an external electric field, even when spin-orbit interaction is absent. The spin-electric coupling can be viewed as originating from a modified exchange constant delta J induced by the electric field. We find that the calculated transition rate between the chiral states yields an effective electric dipole moment d = 3.38 x 10(-33) C m approximate to e10(-4)a, where a is the Cu separation. For external electric fields epsilon approximate to 10(8) V/m this value corresponds to a Rabi time tau approximate to 1 ns and to a delta J on the order of a few mu eV.

• 6.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Dept. of physics, Linnéuniversitetet.
Nordita, Sweden. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
Impurity potential induced gap at the Dirac point of topological insulators with in-plane magnetization2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 15, p. 1-6, article id 155401Article in journal (Refereed)

The quantum anomalous Hall effect (QAHE), characterized by dissipationless quantized edge transport, relies crucially on a nontrivial topology of the electronic bulk band structure and a robust ferromagnetic order that breaks time-reversal symmetry. Magnetically doped topological insulators (TIs) satisfy both these criteria, and are the most promising quantum materials for realizing the QAHE. Because the spin of the surface electrons aligns along the direction of the magnetic-impurity exchange field, only magnetic TIs with an out-of-plane magnetization are thought to open a gap at the Dirac point (DP) of the surface states, resulting in the QAHE. Using a continuum model supported by atomistic tight-binding and first-principles calculations of transition-metal doped Bi2Se3, we show that a surface-impurity potential generates an additional effective magnetic field which spin polarizes the surface electrons along the direction perpendicular to the surface. The predicted gap-opening mechanism results from the interplay of this additional field and the in-plane magnetization that shifts the position of the DP away from the Γ point. This effect is similar to the one originating from the hexagonal warping correction of the band structure but is one order of magnitude stronger. Our calculations show that in a doped TI with in-plane magnetization the impurity-potential-induced gap at the DP is comparable to the one opened by an out-of-plane magnetization.

• 7.
Cent Michigan Univ, USA.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Univ Texas El Paso, USA.
A multiferroic molecular magnetic qubit2019In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 151, no 17, p. 1-7, article id 174105Article in journal (Refereed)

The chiral Fe3O(NC5H5)3(O2CC6H5)6 molecular cation, with C-3 symmetry, is composed of three six-fold coordinated spin-carrying Fe3+ cations that form a perfect equilateral triangle. Experimental reports demonstrating the spin-electric effect in this system also identify the presence of a magnetic uniaxis and suggest that this molecule may be a good candidate for an externally controllable molecular qubit. Here, we demonstrate, using standard density-functional methods, that the spin-electric behavior of this molecule could be even more interesting as there are energetically competitive reference states associated with both high and low local spins (S = 5/2 vs S = 1/2) on the Fe3+ ions. Each of these structures allow for spin-electric ground states. We find that qualitative differences in the broadening of the Fe(2s) and O(1s) core levels, shifts in the core-level energies, and the magnetic signatures of the single-spin anisotropy Hamiltonian may be used to confirm whether a transition between a high-spin manifold and a low spin manifold occurs.

• 8.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics. Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
As vacancies in MnGaAs: tight binding and first-principles studies2012Conference paper (Refereed)
• 9.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
Virginia Commonwealth Univ, Richmond, VA 23284 USA. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
Electronic structure and magnetic properties of Mn and Fe impurities near the GaAs (110) surface2014In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 89, no 16, p. Article ID: 165408-Article in journal (Refereed)

Combining density functional theory calculations and microscopic tight-binding models, we investigate theoretically theelectronic and magnetic properties of individual substitutional transition-metal impurities (Mn and Fe) positioned in the vicinity of the (110) surface of GaAs. For the case of the [Mn2+](0) plus acceptor-hole (h) complex, the results of a tight-binding model including explicitly the impurity d electrons are in good agreement with approaches that treat the spin ofthe impurity as an effective classical vector. For the case of Fe, where both the neutral isoelectronic [Fe3+](0) and the ionized [Fe2+](-)states are relevant to address scanning tunneling microscopy (STM) experiments, the inclusion of d orbitals is essential. We find that the in-gap electronic structure of Fe impurities is significantly modified by surface effects. For the neutral acceptor state [Fe2+, h](0), the magnetic-anisotropy dependence on the impurity sublayer resembles the case of [Mn2+, h](0). In contrast, for [Fe3+](0) electronic configuration the magnetic anisotropy behaves differently and it is considerably smaller. For this state we predict that it is possible to manipulate the Fe moment, e. g., by an external magnetic field, with detectable consequences in the local density of states probed by STM.

• 10.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Department of Physics, University of Texas at Austin, U.S.A.
Theoretical studies of single magnetic impurities on the surface of semiconductors and topological insulators2013In: MRS Online Proceedings Library/Volume 1564/2013, Materials Research Society, 2013Conference paper (Refereed)

We present results of theoretical studies of transition metal dopants in GaAs, based on microscopic tight-binding model and ab-initio calculations. We focus in particular on how the vicinity of surface affects the properties of the hole-acceptor state, its magnetic anisotropy and its magnetic coupling to the magnetic dopant.  In agreement with STM experiments, Mn substitutional dopants on the (110) GaAs surface give rise to a deep acceptor state, whose wavefunction is localized around the Mn center. We discuss a refinement of the theory that introduces explicitly the d-levels for the TM dopant. The explicit inclusion of d-levels is particularly important for addressing recent STM experiments on substitutional Fe in GaAs. In the second part of the paper we discuss an analogous investigation of single dopants in Bi2Se3 three-dimensional topological insulators, focusing in particular on how substitutional impurities positioned on the surface affect the electronic structure in the gap.  We present explicit results for BiSe antisite defects and compare with STM experiments.

• 11.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Virginia Commonwealth University, USA. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
Interplay between Mn-acceptor state and Dirac surface states in Mn-doped Bi2Se3 topological insulator2014In: MAR14 Meeting of The American Physical Society, Denver, Colorado: American Physical Society , 2014Conference paper (Refereed)
• 12.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
Interplay between Mn-acceptor state and Dirac surface states in Mn-doped Bi2Se3 topological insulator2014In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 90, p. Article ID: 195441-Article in journal (Refereed)

We investigate the properties of a single substitutional Mn impurity and its associated acceptor state on the (111) surface of Bi$_2$Se$_3$ topological insulator. Combining \textit{ab initio} calculations with microscopic tight-binding modeling, we identify the effects of inversion-symmetry and time-reversal-symmetry breaking on the electronic states in the vicinity of the Dirac point. In agreement with experiments, we find evidence that the Mn ion is in ${+2}$ valence state and introduces an acceptor in the bulk band gap. The Mn-acceptor has predominantly $p$--character, and is localized mainly around the Mn impurity and its nearest-neighbor Se atoms. Its electronic structure and spin-polarization are determined by the hybridization between the Mn $d$--levels and the $p$--levels of surrounding Se atoms, which is strongly affected by electronic correlations at the Mn site. The opening of the gap at the Dirac point depends crucially on the quasi-resonant coupling and the strong real-space overlap between the spin-chiral surface states and the mid-gap spin-polarized Mn-acceptor states.

• 13.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
The role of d levels of substitutional magnetic impurities at the (110) GaAs surface2013Conference paper (Other academic)

The study of the spin of individual transition-metal dopants in a semiconductor host is an emergent field known as magnetic solotronics, bearing exciting prospects for novel spintronics devices at the atomic scale. Advances in different STM based techniques allowed experimentalists to investigate substitutional dopants at a semiconductor surface with unprecedented accuracy and degree of details [1]. Theoretical studies based both on microscopic tight-binding (TB) models and DFT techniques have contributed in elucidating the experimental findings. In particular, for the case of Mn dopants on the (110) GaAs surface, TB models [2] have provided a quantitative description of the properties of the associated acceptor states. Most of these TB calculations ignore dealing explicitly with the Mn d-levels and treat the associated magnetic moment as a classical vector. However recent STM experiments [3] involving other TM impurities, such as Fe, reveal topographic features that might be related to electronic transitions within the d-level shell of the dopant. In this work we have included explicitly the d levels in the Hamiltonian. The parameters of the model have been extracted from DFT calculations. We have investigated the role that d levels play on the properties of the acceptor states of the doped GaAs(110) surface, and analyzed their implications for STM spectroscopy.

• 14.
Virginia Commonwealth University, USA.
Virginia Commonwealth University, USA. Virginia Commonwealth University, USA. Virginia Commonwealth University, USA.
Robust Magnetic Moments on Impurities in Metallic Clusters: Localized Magnetic States in Superatoms2013In: Journal of Physical Chemistry A, ISSN 1089-5639, E-ISSN 1520-5215, Vol. 117, no 20, p. 4297-4303Article in journal (Refereed)

Introducing magnetic impurities into clusters of simplemetals can create localized states for higher angular momentum quantumnumbers (l = 2 or 3) that can breed magnetism analogous to that invirtual bound states in metallic hosts, offering a new recipe for magneticsuperatoms. In this work, we demonstrate that MnCan clusterscontaining 6−15 Ca atoms show a spin magnetic moment of 5.0 μBirrespective of the cluster size. Theoretical analysis reveals that the Mn dstates hybridize only partially with superatomic states and introduce extramajority and minority d states, largely localized at the Mn site, with alarge gap. Successive addition of Ca atoms introduces superatomic statesof varying angular momentum that are embedded in this gap, allowingcontrol over the stability of the motifs without altering the moment. Assemblies of such clusters can offer novel electronic features due to theformation of localized magnetic “quasibound states” in a confined nearlyfree electron gas.

• 15.
Purdue University, USA.
University of Texas, USA.
Quantum Percolation in Two Dimensions2009In: Quantum and Semi-classical Percolation and Breakdown in Disordered Solids / [ed] Bikas K. Chakrabarti, Kamal K. Bardhan, Asok K. Sen, Springer, 2009Chapter in book (Other academic)

We overview the recent work on the transport of a quantum particle through a two-dimensional, disordered network produced by the percolation algorithm. This problem, known as quantum percolation, has been controversial regarding the nature of localization and presence or absence of phase transitions in its transmission behavior.

• 16.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Lunds universitet.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. US DOE, Off Basic Energy Sci, Washington, DC 20585 USA.
Electric control of a {Fe4} single-molecule magnet in a single-electron transistor2013In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 88, no 22, p. Article ID: 224423-Article in journal (Refereed)

Using first-principles methods, we study theoretically the properties of an individual {Fe-4} single-molecule magnet (SMM) attached to metallic leads in a single-electron transistor geometry. We show that the conductive leads do not affect the spin ordering and magnetic anisotropy of the neutral SMM. On the other hand, the leads have a strong effect on the anisotropy of the charged states of the molecule, which are probed in Coulomb blockade transport. Furthermore, we demonstrate that an external electric potential, modeling a gate electrode, can be used to manipulate the magnetic properties of the system. For a charged molecule, by localizing the extra charge with the gate voltage closer to the magnetic core, the anisotropy magnitude and spin ordering converges to the values found for the isolated {Fe-4} SMM. We compare these findings with the results of recent quantum transport experiments in three-terminal devices.

• 17.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics. Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics. Naval Research Laboratory, Washington DC.
DFT calculations of the charged states of N@C60 and {\{}Fe4{\}} single molecule magnets investigated in tunneling spectroscopy2012Conference paper (Other academic)

For device applications of single molecule magnets (SMMs) in high-density information storage and quantum-state control it is essential that the magnetic properties of the molecules remain stable under the influence of metallic contacts or surface environment. Recent tunneling experiments [1, 2] on N@C60 and {\{}Fe4{\}} SMM have shown that these molecules preserve their magnetic characteristics when they are used as the central island of single-electron transistors. Although quantum spin models have been used extensively to study theoretically tunneling spectroscopy of SMMs, it has been shown recently that the orbital degrees of freedom, which is absent in spin models, can significantly affect the tunneling conductance [3]. In this work we present first-principles calculations of the neutral and charged states of N@C60 and {\{}Fe4{\}} SMMs, and discuss a strategy to include their properties into a theory of quantum transport. We also present results of the magnetic anisotropy for the different charge states of Fe4 and discuss their relevance for experiments [2] in the sequential tunneling and cotunnelling regimes.

• 18.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics. Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics. US Department of Energy SC22.1, Washington DC 20585-1290.
First-principles studies of spin-orbit and Dzyaloshinskii-Moriya interactions in the {Cu3} single-molecule magnet2012In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 85, no 8, p. 085427-1-085427-10Article in journal (Refereed)

Frustrated triangular molecule magnets such as {Cu3} are characterized by two degenerate S = 1/2 ground states with opposite chirality. Recently, it has been proposed theoretically [M. Trif et al., Phys. Rev. Lett. 101, 217201 (2008)] and verified by ab initio calculations [M. F. Islam et al., Phys. Rev. B 82, 155446 (2010)] that an external electric field can efficiently couple these two chiral spin states, even in the absence of spin-orbit interaction (SOI). The SOI is, nevertheless, important since it introduces a splitting in the ground-state manifold via the Dzyaloshinskii-Moriya (DM) interaction. In this paper, we present a theoretical study of the effect of the SOI on the chiral states within spin-density functional theory. We employ a recently introduced Hubbard-model approach to elucidate the connection between the SOI and the Dzyaloshinskii-Moriya interaction. This allows us to express the Dzyaloshinskii-Moriya interaction constant D in terms of the microscopic Hubbard-model parameters, which we calculate from first principles. The small splitting that we find for the {Cu3} chiral state energies (≈ 0.02 meV) is consistent with experimental results. The one-band Hubbard-model approach adopted and analyzed here also yields a better estimate of the isotropic exchange constant than the ones obtained by comparing total energies of different spin configurations. The method used here for calculating the DM interaction unmasks its simple fundamental origin, which is the off-diagonal spin-orbit interaction between the generally multireference vacuum state and single-electron excitations out of those states.

• 19.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics. Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Spin-orbit interaction and magnetic field in antiferromagnetic triangular molecular magnets2011In: APS MARCH MEETING 2011, 2011Conference paper (Refereed)
• 20.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Virginia Commonwealth University, USA. Department of Physics, University of Texas at Austin, U.S.A..
Theoretical studies of surface states in Bi2Se3: effects of finite thickness, finite-cluster boundaries, and surface doping2013Conference paper (Refereed)

Recently, a family of bismuth-based materials, in particular bismuth chalcogenides (Bi2Se3, Bi2Te3), have been identified as three-dimensional (3D) topological insulators (TIs), i.e. materials characterized by a non-trivial bulk insulating gap and topologically protected surface states with linear dispersion and helical spin texture, traversing the gap [1].  A question of great fundamental and practical importance is how electronic and spin properties of topological surface states are modified in the presence of external perturbations, in particular time-reversal-breaking ones, such as magnetic dopants [2]. Experimentally, this question can be addressed by using advanced experimental probes, such as spin-sensitive angle-resolved photoemission spectroscopy (ARPES) and scanning tunnelling microscopy (STM). On the theoretical side, there is a need for atomistic modelling of TIs that enables quantitative analysis and comparison with experiments, while keeping the computational overhead to a minimum. Microscopic tight-binding models, combined with input from ab initio calculations, provide a convenient platform to study surface states in TIs [3].

We present results of realistic tight-binding modelling of 3D TIs, with particular focus on Bi2Se3. Our implementation is based on the sp3 tight-binding model for Bi2Se3 by Kobayashi [4], with parameters calculated from density functional theory. We start with a thorough analysis of the calculated band structure of a slab of Bi2Se3 of varying thickness, with surface states identified using quantitative criteria according to their actual spatial distribution. We investigate the thickness-dependent energy gap for thin slabs, attributed to inter-surface interaction, and the emergence of gapless surface states for slabs above a critical thickness. A quantitative description of the transition to infinite-thickness limit is provided by calculating explicitly the associated modifications in the spatial distribution and spin character of the wave function. We find that the system must be at least forty quintuple-layers thick to displace surface states that are essentially localized on either surface. In addition, we discuss the effect of an external magnetic field on the electronic structure of Bi2Se3. The peculiar structure of the Landau levels, found experimentally in this system [5], is a characteristic signature of the presence of Dirac surface states and can be used to extract the dispersion of the surface band. Furthermore, building upon previous work on GaAs [6], we develop a finite-cluster tight-binding approach, where an infinite slab is represented by a large but finite cluster with the same thickness. We find that the electronic structure for a finite cluster, with periodic boundary conditions along the length and width of the cluster, is in excellent agreement with that of an infinite slab. On the other hand, the finite-cluster approach allows us to directly probe mesoscopic effects, associated with real boundaries of a finite system, especially when the topological surface states are present. The approach enables also the investigation of individual impurities and defects. As a first application we present a case study of substitutional Bi defect at Se site near the surface of a slab of Bi2Se3 of varying thickness, focusing in particular on the interplay between defect-induced states and gapless surface states.  The analysis of spatial features of the calculated local density of states around the defect reveals similarities with STM studies [7]. Finally, we discuss strategies for incorporating single transition-metal dopants (Mn, Fe) in the current model and draw connections to recent STM experiments [8].

[1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010); X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).

[2] H. Beidenkopf et al., Nature Physics 7, 939 (2011); L. A. Wray et al., Nature Physics 7, 21 (2011).

[3] W. Zhang, R. Yu, H.-J. Zhang, X. Dai, and Z. Fang, New. J. Phys. 12, 065013 (2010); M. S. Bahramy et al., Nature Communications 3, 1159 (2012).

[4] K. Kobayashi, Phys. Rev. B 84, 205424 (2011).

[5] T. Hanaguri et al., Phys. Rev. B 82, 081305(R) (2010).

[6] T. O. Strandberg et al., Phys. Rev. B 80, 024425 (2009).

[7] S. Urazhdin et al., Phys. Rev. B 66, 161306(R) (2002); S. Urazhdin et al., Phys. Rev. B 69, 085313 (2004).

[8] Y. S. Hor et al., Phys. Rev. B 81, 195203 (2010);  T. Schlenk et al., Phys. Rev. Lett. 110, 126804 (2013).

• 21.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Lund University.
Polish Academy of Sciences, Poland. Polish Academy of Sciences, Poland. Polish Academy of Sciences, Poland. Polish Academy of Sciences, Poland. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering. Polish Academy of Sciences, Poland.
Wurtzite (Ga,Mn)As nanowire shells with ferromagnetic properties2017In: Nanoscale, ISSN 2040-3364, E-ISSN 2040-3372, Vol. 9, no 6, p. 2129-2137Article in journal (Refereed)

(Ga,Mn)As having a wurtzite crystal structure was coherently grown by molecular beam epitaxy on the 1100 side facets of wurtzite (Ga,In)As nanowires and further encapsulated by (Ga,Al)As and low temperature GaAs. For the first time, a truly long-range ferromagnetic magnetic order is observed in non-planar (Ga,Mn)As, which is attributed to a more effective hole confinement in the shell containing Mn by the proper selection/choice of both the core and outer shell materials. Â© The Royal Society of Chemistry.

• 22.
Virginia Commonwealth University.
Virginia Commonwealth University. Virginia Commonwealth University.
Using graphene to control magnetic anisotropy and interaction between supported clusters2015In: New Journal of Physics, ISSN 1367-2630, E-ISSN 1367-2630, Vol. 17, article id 053052Article in journal (Refereed)

Stabilization of magnetic order in clusters/nanoparticles at elevated temperatures is a fundamentallychallenging problem. The magnetic anisotropy energy (MAE) that prevents the thermal fluctuationsof the magnetization direction can be around 1–10Kin free transition metal clusters of around adozen atoms. Here we demonstrate that a graphene support can lead to an order of magnitudeenhancement in the anisotropy of supported species. Our studies show that theMAEof supportedCo5 and Co13 clusters on graphene increase by factors of 2.6 and 25, respectively. The enhancement islinked to the splitting of selected electronic orbitals that leads to the different orbital contributionsalong the easy and hard axis. The conductive support enables a magnetic interaction between thedeposited species and the nature of themagnetic interaction can be controlled by the separation betweensupported clusters or by vacancies offering an unprecedented ability to tune characteristics of assemblies.

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Opuscula Math. 28, no. 4 (2008), 529-539

Opuscula Mathematica

# Iteration groups, commuting functions and simultaneous systems of linear functional equations

Janusz Matkowski

Abstract. Let $$( f^t )_{t \in \mathbb{R}}$$ be a measurable iteration group on an open interval $$I$$. Under some conditions, we prove that the inequalies $$g\circ f^a \leq f^a \circ g$$ and $$g\circ f^b \leq f^b\circ g$$ for some $$a,b \in \mathbb{R}$$ imply that $$g$$ must belong to the iteration group. Some weak conditions under which two iteration groups have to consist of the same elements are given. An extension theorem of a local solution of a simultaneous system of iterative linear functional equations is presented and applied to prove that, under some conditions, if a function $$g$$ commutes in a neighbourhood of $$f$$ with two suitably chosen elements $$f^a$$ and $$f^b$$ of an iteration group of $$f$$ then, in this neighbourhood, $$g$$ coincides with an element of the iteration group. Some weak conditions ensuring equality of iteration groups are considered.

Keywords: iteration group, commuting functions, functional equation, functional inequalities.

Mathematics Subject Classification: 39B12, 39B72, 26A18.

Full text (pdf)

• Janusz Matkowski
• University of Zielona Góra, Institute of Mathematics, Computer Science and Econometry, ul. Podgórna 50, 65-246 Zielona Góra, Poland
• Silesian University, Institute of Mathematics, 40-007 Katowice, Poland
• Revised: 2008-02-10.
• Accepted: 2008-01-20.

format: plain


text:
### Decorate the wall

Time Limit: 1000 ms Memory Limit: 65536 KiB

#### Problem Description

After building his huge villa, Mr. Rich cannot help but notice that the interior walls look rather blank. To change that, he starts to hang paintings from his wonderful collection. But soon he realizes that it becomes quite difficult to find a place on the wall where a painting can be placed without overlapping other paintings. Now he needs a program which would tell him, given the already placed paintings, where to place the next painting without moving any other paintings (or indicating that this is impossible). Paintings have a rectangular shape and are to be placed parallel to the side of the wall. If you do not mind a nice reward from Mr. Rich, go on and solve the problem.

#### Input

The first line of the input file contains a number representing the number of test cases to follow. Each test case starts with a line containing three numbers n, w and h. n is the number of paintings already hanging on the wall, w is the width of the wall and h is the height of the wall. The next n lines contain 4 integers x1, y1, x2, y2 each (0 ≤ x1< x2 ≤ w, 0 ≤ y1 < y2 ≤ h); the x-coordinates give the distance to the left end of the wall, the y-coordinates give the distance to the bottom of the wall. (x1, y1) is the position of the lower left corner of a painting, (x2, y2) is the position of the upper right corner. The last line of each test case contains the dimensions of the next painting to be placed, first its width w\', then its height h\' (1 ≤ w\' ≤ w, 1 ≤ h\' ≤ h). You are not allowed to rotate the painting.
You can assume that 0 ≤ n ≤ 200 and 1 ≤ w, h ≤ 1000000. Moreover, all paintings already hanging do not overlap.

#### Output

Produce one line of output for each test case. Write "Fail!" if there is no place left on the wall where the painting could be placed without overlapping other paintings. Otherwise, write the coordinates where the lower left corner of the painting should be placed. In case there is more than one solution, select the solution with a minimum y-coordinate, and break ties using the minimum x-coordinate.

#### Sample Input

2
1 10 9
5 4 10 9
9 5
2 10 10
5 5 10 10
0 0 4 3
3 4

#### Sample Output

Fail!
4 0

#### Source

2005~2006 University of Ulm Local Contest


text:
October 25, 2019

# Exponentially Growing Learning Rate? Implications of Scale Invariance induced by Batch Normalization

Yesterday I read this intriguing paper about the midboggling fact that it is possible to use exponentially growing learning rate schedule when training neural networks with batch normalization:

The paper provides both theoretical insights as well as empirical demonstration of this remarcable property.

## Scale invariance

The reason why this works boils down to the observation that batch-normalization renders the loss function of neural networks scale invariant - scaling the weights by a constant does not change the output, or the loss, of the batch normalized network. It turns out that this property alone might result in somewhat unexpected and potentially helpful properties for optimization. I will use this post to illustrate some of the properties of scale invariant loss functions - and gradient descent trajectories on them - using a 2D toy example:

Here, I drew a loss function whih has the scale invariance property. The value of the loss only depends on the angle, but not the magnitude of the weight vector. The value of the loss along any radial line from the origin outwards is constant. Simple consequences of scale invariance are that (Lemma 1 of the paper)

1. that the gradient of this function is always orthogonal to the current value of the parameter vector, and that
2. the farther you are from the origin, the smaller the magnitude of the gradient. This is perhaps less intuitive but think about how the function behaves on a circle around the origin. The function is the same, but as you increase the radius you stretch the same function round a larger circle - it gets fatter, therefore its gradients decrease.

Here is a - somewhat messy - quiver plot showing the gradients of the function above:

The quiver plot is messy because the gradients around the origin explode. But you can perhaps see how the gradients get larger and larger - and remain perpendicular to the value itself.

So Imagine doing vanilla gradient descent (no momentum, weight decay, fixed learning rate) on such a loss surface. Because the gradient is always perpendicular to the current value of the parameter, by the Pythagorean theorem, the norm of the parameter vector increases with each iteration. So gradient descent takes you away from the origin. However, the weight vector won't completely blow up to infinity, because the gradients also get smaller and smaller as the weight vector grows, so it settles at some point. Here is a gradient descent path looks like starting from the coordinate $(-0.7, 0.7)$:

In fact, you can't really see it but the optimization kind of gets stuck in there, and doesn't move any longer. It's interesting to see what happens if we add weight decay, which is the same as adding L2 regularizer over the weights:

We can see that once the trajectory is about to get stuck in a local minimum, weight decay pulls it back towards the origin, which is where gradients become larger. This, in turn, perturbs the trajectory often pushing it out of the current local minimum. So in a way, we can start to build the intuition that weight decay on a scale-invariant loss function acts as a kind of learning rate adjustment.

In fact, what the paper works out is an equivalence between two things:

• weight decay with constant learning rate and
• no weight decay and an exponentially growing learning rate

On the plot below I show the trajectory with the exponentially growing learning rate which is equivalent to the one I showed before with weight decay. This one has no weight decay, and its learning rate keeps growing:

We can see that the trajectory blows up, and quickly gets out of bound on this animation. How can this be equivalent to the weight decay trajectory? Well, from the perspective of the loss function, the magnitude of the weight vector is irrelevant, and we only care about the angle when viewed from the origin. Turns out, if you look at those angles, the two trajectories are the same. To illustrate this, I use the normalization formula from Theorem 2.1 to project this trajectory back to the same magnitude the weight decay one would have. I obtain something that indeed looks very much like the trajectory above:

After a while, the trajectories start working differently, which I think is probably due to the accumulation of numerical errors in my implementation of the toy example. I could probably fix this, but I'm not sure it's worth the effort. The authors show much more convincing empirical evidence that this works in real, complicated neural network losses that people actually want to optimize.

You can think of this renormalization I did above as "constantly zooming out" on the loss landscape to keep up with the exponentially exploding parameter. I tried to illustrate this below:

On the left-hand plot, I show the original, weight-decayed gradient descent with a constant learning rate. On the right-hand plot I show the equivalent trajectory with exponentially growing learning rate and no weight decay, and I also added a constant zoom to counteract the explosion of the parameter's norm, in line with Theorem 2.1. We can see that, especially initially, the two paths behave the same way when viewed from the origin. They then work differently which I believe is down to the numerical precision issue that could probably be worked out.

The paper shows a similar equivalence in the presence of momentum as well, if interested, read the details in the paper.

## Summary

I thought this observation was very cool, and may well lead to a better understanding of the mechanisms by which batchnorm and other weight normalization schemes work. It also explains why the combination of weight decay with weight normalization schemes results in a relatively robust gradient descent regime where constant learning rate works well.


text:
I pull back the string on a pendulum until it is at 60 degrees. I let go of it. When the pendulum is at its minimum point (theta=0), what is its speed?

Possibly useful info: The string is 0.4 meters long.also m=0.05 km

Give your answer in units of meters per second, BUT DO NOT PUT THE UNITS IN YOUR ANSWER, JUST GIVE THE NUMBER THAT YOU GET IF YOU DO YOUR CALCULATION IN METERS PER SECOND!

part b
If I had asked you for the speed when it reaches theta=-30 degrees (i.e. it's on the left) what would your answer to the previous part be multiplied by?


text:
# Mann Whitney Wilcoxon test [duplicate]

My stats background is not very strong so I was hoping I'd be able to get some help. In my MSc thesis I am using R to test for significant differences in means between populations. My data is not normally distributed (based on QQplots, Shapiro-Wilk's and histograms), therefore I cannot use parametric tests. So I am using a Mann Whitney Wilcoxon test instead (wilcox.test in R).

Now from my understanding Mann Whitney Wilcoxon test for significant differences in medians, not means. I have a table with my data reported as means, but I report significance testing results in my thesis based on medians as calculated by R. Is this an appropriate approach? Or do I need to convert the means to medians in my table?

Thanks.

## marked as duplicate by Scortchi - Reinstate Monica♦Aug 9 at 6:59

• It doesn't test for difference in medians. It corresponds to a test for the median pairwise difference (in the population). You can get a rejection when the sample medians are equal. Many posts on site discuss this issue. – Glen_b Mar 18 '18 at 9:37

The Wilcoxon test is acutally not a test of medians but of ranks. There is a median test, but that apparently has low power(1) ist thus rarely used (R obviously has a package - you know! (2)). You might also want to look in the wilcox.test- help pages to read about a "pseudomedian": help(wilcox.test)

There is no logical problem with presenting means whilst proving difference in ranks. You may consider it "not elegant", if you are so inclined. There is no problem in reporting the mean as well as the median, if that helps.

Chances are, that you can actually use parametric tests in the absence of normality, if you have enough data. It is hard to tell, how much data is enough for how big a deviance from normality. Some say $n>30$ or $n>50$ might bie good rules of thumb.

And finally, if you have enough observations, you might compare means of non-normal data by bootstrapping...

• Thank you for your comment. In my methods I said " Non-parametric Mann Whitney Wilcoxon test was used to test for differences in medians..", so I guess I should take out the medians from that sentence? – chrisuw92 Mar 14 '18 at 7:00
• Yes, you should take it out, as it is wrong. The median is a common way to report the same kind of data that the Wilcoxon tests are used to examine, which makes this a common misconception. You probably do not want to know about the median, you want to know, whether values in one group are systematically higher then in the other group. That, the Wilcoxon tests do. – Bernhard Mar 14 '18 at 7:21
• The test is actually about testing equality of distributions with respect to location based on ranks. It is not a test of ranks. The idea is that if the distributions are shifted significantly the low ranks would tend to come mostly from distribution and the high ranks from distribution B. If the distributions are the same the ranks would be intertwined and so the low ranks and the high ranks would tend to come "equally from both A and B. – Michael Chernick Mar 14 '18 at 9:25
• If the distributions are shifted and the means exist the test would indicate both a difference in means and medians. – Michael Chernick Mar 14 '18 at 9:27

Mann-Whitney only tests for a difference in medians when you assume that the only difference in the distributions of the two samples is the location, and not the scale or shape, of the distribution, which is very often too strong an assumption.

In fact, if you just want a test for a difference in means (which is often the case), t-testing will work just fine even if the data aren't normal provided that you have a reasonable sample size, as Bernhard mentioned.

• (+1) But note that under the location-shift assumption, a test for a difference in medians is a test for a difference in means. – Scortchi - Reinstate Monica Aug 9 at 7:04
• With large samples a t-test will (usually) work okay as far as significance level goes. Power can sometimes be a different matter. – Glen_b Sep 23 at 5:08


text:
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text:
A note on a problem of S\''{a}rk\"{o}zy and S\''{o}s
Received:February 20, 2021  Revised:February 20, 2021
Key Word: representation function   linear form
Fund ProjectL:
 Author Name Affiliation Address Min Tang School of Mathematics and Statistics, Anhui Normal University School of Mathematics and Computer Science，Anhui Normal University
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Abstract:
Let $k,\ell \geq 2$ be positive integers. Let $A$ be an infinite set of nonnegative integers. For $n\in \mathbb{N}$, let $r_{1,k,\ldots,k^{\ell-1}} (A, n)$ denote the number of solutions of $n=a_0+ka_1+\cdots +k^{\ell-1}a_{\ell-1}$, $a_0, \ldots, a_{\ell-1}\in A$. In this paper, we show that $r_{1,k,\ldots,k^{\ell-1}} (A, n)=1$ for all $n\geq 0$ if and only if $A$ is the set of all nonnegative integers such that all its digits in its $k^\ell$-adic expansion are smaller than $k$. This result partially answers a question of S\''{a}rk\"{o}zy and S\''{o}s on representation for multivariate linear forms.
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# inequality 2

Algebra Level 1

What is the least possible value of $$\frac{9}{1+2y^2}$$ if $$0 \leq y \leq 2$$

×


text:
## Brown University: Rich Schwartz's "Countable and Uncountable Sets"

Read this document to learn about countable and uncountable sets. Focus on the several examples of uncountable subsets of R.


text:
What is the cause for the difference in pH between NaCl and Ringer Lactate

"Normal saline contains approximately $$\pu{154mM}$$ of each $$\ce{Na+}$$ and $$\ce{Cl-}$$ with average pH of 5.0 and osmolarity of $$\pu{308mOsm/L}$$. LR solution has electrolytes $$\pu{130mM}$$ of $$\ce{Na+}$$, $$\pu{109mM}$$ of $$\ce{Cl-}$$, and $$\pu{28mM}$$ of lactate. The Average pH of LR is 6.5 and is slightly hypo-osmolar with $$\pu{272mOsm/L}$$. 6,7"

Regular $$\ce{NaCl}$$ solutions should have 7 as a pH, yet this solution is 5. A lactate ringer which has a similar composition to regular $$\ce{NaCl}$$ is 6.5.

I thought that the pH of 5 doesn't have anything to do with the $$\ce{NaCl}$$ but with the reaction of blood $$\ce{CO2}$$ with the water in the $$\ce{NaCl}$$ solution lowering the pH, but clearly that was wrong.

• Note that pH of water in equilibrium with air is expected to be about 5.6. NaCl itself does not cause pH lowering, but CO2 and possible impurities do. – Poutnik Jun 15 at 6:03
• Note that presence of lactate as salt of weak acid breaks the similarity from acid-base equilibrium point of view. – Poutnik Jun 16 at 14:45


text:
How to prevent Signal drifting when changing frequency

When I recreate a signal it seems to look great. But when I try and increase it's frequency (which is done in a for loop and the variable new_freq=2) it starts to drift.

See Image below (The top image is 1hz and the bottom image is 2hz. I circled the image with the problem). I also noticed when I increase the sample frequency the problem gets even more noticeable. Notice the top one is at 8000 as the sample frequency and the bottom one is at sample frequency of 44100.

I've posted the for loop below any ideas why the variables aa_sig_rebuilt_norm_bin and aa_sig_combined_norm_bin are drifting so much? Please note I'm using much more complex signals than just sine waves (if I were just using sine waves I would use the sinwave equation to increase the frequency. I used a sinewave as an example)

The whole for loop is base on rebuilding a signal using the equation AmplitudeCos(freqtime+phase) I did an fft/ifft to get all the frequencies/amplitudes/phases then just change the frequency to what I want

I do want to vectorize this for loop but at the moment I'm trying to fix the drift

new_freq=[2]; %increase frequency
fn_array=[];

t_rebuilt=linspace(0,2*pi,fs);

for zz=1:1:length(new_freq); %

zz
%needs to be in for loop to reset variables for next new_freq
aa_sig_rebuilt_norm=zeros(1,length(t_rebuilt));
aa_sig_rebuilt_norm_bin=zeros(1,length(t_rebuilt));
aa_sig_combined_norm=zeros(1,length(t_rebuilt));
aa_sig_combined_norm_bin=zeros(1,length(t_rebuilt));
sig_norm=zeros(1,length(t_rebuilt));
sig_norm_bin=zeros(1,length(t_rebuilt));
for kk=1:1:length(xfreq_orig);
aa_sig_rebuilt_norm = array1(kk, 2)*cos (array1(kk,1)*t_rebuilt+(array1(kk, 4)));
aa_sig_combined_norm = aa_sig_combined_norm + aa_sig_rebuilt_norm;

aa_sig_rebuilt_norm_bin = array1(kk, 2)*cos ((new_freq(zz,1))*array1(kk,1)*t_rebuilt+(array1(kk, 4))); %binural section
aa_sig_combined_norm_bin = aa_sig_combined_norm_bin + aa_sig_rebuilt_norm_bin;
end;

%have it export to file
sig_norm=(aa_sig_combined_norm/max(abs(aa_sig_combined_norm))*.8); %normalize signal
sig_norm_bin=(aa_sig_combined_norm_bin/max(abs(aa_sig_combined_norm_bin))*.8); %normalize signal

wavwrite([sig_norm' sig_norm_bin'] ,fs,16,strcat(dirpathtmp,fn_LR_norm_bin)); %export file

end;

• This is a programming question and not a signal processing question. I vote to close it as off-topic for this site. – Dilip Sarwate Apr 27 '13 at 11:55
• What do you mean by "increase it's frequency with a for loop?" What are you actually trying to do to the signal? It's not clear from a brief look at your code dump. It would be most useful if you provided some more details including a self-contained runnable example. – Jason R Apr 27 '13 at 11:55
• @Jason R when I mean "increase frequency with a for loop" I should have just said increase frequency. So if a vocal signal is defined as 1hz if I put a 2 in the variable new_freq=[2] then it will double it's frequency. The code uses multiple functions and one guy already wants to close this question for posting a small for loop that deals with creating the signal and pictures to help explain things why, I'm not sure, so adding more code will just piss him off even more. – Rick T Apr 27 '13 at 12:09
• @RickT: Dilip brings up a good point. Your question as currently posed isn't a very good one. Instead of asking how to fix your implementation, a question more along the lines of how to appropriately shift the pitch of a voice signal would be better. – Jason R Apr 27 '13 at 16:40
• I don't have an answer for that. You would be better off posting a new question that is based around that query, as it will garner more attention by those who have expertise in that area. – Jason R Apr 27 '13 at 17:44


text:
Our Law of Sines Calculator comes in help when working out problems with angles or side lengths. We’ll go through the law of sines formula and offer you a list of situations when this rule might come in handy. You may now answer various trigonometry problems with the help of this calculator (more elaborate than the Pythagorean theorem).

This calculator calculates unknown values of a figure using the law of sines. You must input three known values in order to compute the unknown values. Some of the calculation options are redundant, but they are still supplied for accurate letter designations. If you want to know how to find the gradients and margins of a triangle, please check out our Triangle Angle Calculator of geometry, it can be very useful in solving triangles.

## Law of Sines

The law of sines provides the math link between an oblique triangle’s sides and angles (non-right triangle). The laws of sines and the law of cosines in trigonometry are key principles for solving a triangle. The law of sines definition (sine rule) states that the ratios of a triangle’s side lengths to the sine of its respective opposing gradients are equal. In the next parts, we’ll look at the formula and how to prove it using solved instances.

The rule connects the ratios of triangle side lengths to their opposing angles. For all three sides and opposing gradients, this ratio stays constant. Using the needed known data, we may use the sine rule to calculate any triangle’s missing gradient or side. The diameter of the circumcircle of one triangle is equal to the ratio of the side and the corresponding angle.

### History

The spherical rule of sines was found in the 10th century, according to Ubiratàn D’Ambrosio and Helaine Selin. Abu-Mahmud Khojandi, Abu al-Wafa’ Buzjani, Nasir al-Din al-Tusi, and Abu Nasr Mansur are all said to have written it. Ibn Mudh al-The Jayyn’s book of unknown arcs of a sphere, written in the 11th century, the general geometry and math law of sines.

Nasr al-Dn al-Ts subsequently articulated the plane rule of sines in the 13th century. He stated the law of sines for flat and spherical triangles in his On the Sector Figure and presented evidence for it. “The Law of Sines is truly Regiomontanus’ basis for his solutions of right-angled triangles in Book IV, and these solutions are in turn the grounds for his solutions of general triangles,” according to Glen Van Brummelen. Regiomontanus was a German geometry mathematician who lived in the 15th century.

Any triangle’s area T equals one half of its base times its height. When one of the triangle’s margins is chosen as the base, the triangle’s height is calculated as the length of another side multiplied by the sine of the gradient between the chosen side and the base.

## Law of Sines Formula

The law of sines formula is utilized to link the lengths of a triangle’s sides to the sines of consecutive angles. It is the ratio of the length of one of the triangle’s sides to the sine of the gradient created by the other two borders.

Apart from the SAS and SSS triangles, the law of sine formula is applied to any triangle. There are so many equations for the law of sines. But that many equations can be very difficult, so we will provide you with the easiest one, for better understanding. So the law of sines equation is:

\frac {a} {\sin(\alpha)} = \frac {b} {\sin(\beta)} = \frac {c} {\sin(\gamma)}

Where:

• a, b, and c are the lengths of the triangle,
• α, β, and γ are angles of the triangle.

Our calculator will provide you with a few types of triangles.

### ASA Triangle

So, suppose any two angles and the included side of one triangle is congruent to two angles and the included side of another triangle. In that case, the two triangles are congruent, according to the Angle-Side-Angle Postulate (ASA). The ASA is used to argue that triangle ABC is congruent to triangle DEF, as seen in the diagram down.

### AAS Triangle

The Angle-Angle-Side Postulate (AAS) states that if two gradients and a non-included side of one triangle are equivalent to two angles and a non-included side of another triangle, the two triangles are congruent. The AAS Postulate demonstrates that triangle ABD is congruent to triangle CBD, as seen in the accompanying illustration.

### SSA Triangle

The acronym SSA stands for side – side – angle. t refers to a triangle having two known lengths and one known angle that is not between the two sides. We can work out the remaining side and the other two angles if we have this much information.

For example, consider a math triangle having area of a, b, and c sides. Angle A is the opposite side of a angle B is the opposing side of b, and angle C is the opposite side of c. We know that side a is 8 inches long and side b is 6 inches long in this triangle. We also know that A is a 50-degree angle.

## Law of Sines Calculator – How to Use?

Begin by defining your problem. You may, for example, know two angles and one side of a triangle and are seeking the other sides. Fill up the blanks on this triangle calculator using the known values. Remember to double-check whether you used the right symbols to represent the margins and angle values.

Given two angles and a side, the law of sine is used to calculate the remaining sides of a triangle. Triangulation is the term for this method. We can also use it when two sides and one of the non-enclosed gradients are specified. However, in other circumstances, such as the ambiguous example, the triangle cannot be uniquely identified by the available data. We get two alternative values for the enclosed angle.

Enter the opposite side b, then another angle-side pair such as A and an or C and c to compute any angle, A, B, or C, say B. The side gradient (SSA) approach is used in the calculations. The law of sine is only used to finish the calculations for other unknowns. Enter the opposite angle B and then another angle-side pair such as A and an or C and c to compute any side, a, b, or c, say b. The calculations are done using the angled side (AAS) approach, and the law is only used to finish the calculations for other unknowns.

Wow, this calculator is really awesome!

## Law of Sines Calculator – Example

Let us now provide you with an example of how to calculate it. So let’s solve this area. First put in the values we know:

• Choose the rule you want to apply – for the sake of example let’s say it’s a/sin(α) = c/sin(γ)
• Input the side a = 7 cm, α = 35 degrees; side c = 10 cm
• Unknown is the γ, and we obtain the result γ = 55,024 degrees

Try out this very useful calculator with your own examples.

## FAQ

How to solve triangles using the law of sines?

Simply expressed, it states that the ratio of the length of a side to the sine of the gradient opposite that side is the same for all borders and gradients of a triangle.

When can we use the Law of Sines?

The rule of sines may be used to compute the remaining sides of a triangle when two angles and a side are known, a procedure known as triangulation. Likewise, it can also be employed when two sides and one of the non-enclosed angles are known.

Does the Law of Sines work for all triangles?

The Sine Rule may be used to any triangle with a known side and opposing angle (not only right-angled triangles). Only two of the three elements of the formula will ever be required. To utilize the Sine Rule, you’ll need to know at least one set of sides and their opposing angle.

How do you find two triangles using the law of sines?

If you want to calculate one of the other two angles, first, use the rule of sine. Then, to obtain the third one, substract the sum of the remaining two from the 180 degrees.

When to use the law of sines?

When given a gradient and two sides, this law of sines may be used to locate a missing angle or a missing side when given two gradients and one side.


text:
# FAQ overflow

#### QUESTION

I want to format a step-by-step tutorial in LaTeX in a way that each step shows one screenshot and a small piece of text for that step. I could think of a formatting where in each step the screenshot is positioned on the left and the text is on the right. However, if I use \includegraphics the image is dynamically positioned in the document.

How can I realize such formatting?

It's by no means necessary to place an \includegraphics command inside a figure environment.

If the text doesn't overflow much the figure's height, you can use a simple method:

\documentclass[a4paper]{article}
\usepackage[demo]{graphicx}
\usepackage{lipsum}

\newenvironment{explanation}[2][]
{\begin{flushleft}
\begin{minipage}[t]{\dimexpr\textwidth-6cm\relax}}
{\end{minipage}\end{flushleft}}

\begin{document}
\lipsum[2]

\begin{explanation}{screenshot1}
This is the text that accompanies the first screen shot
and explains it.
\end{explanation}

\lipsum[2]
\end{document}

Note: the demo option for graphicx is just to allow avoiding having real figure files. Here I've provided 6cm space for the screen shot, which is typeset as 5cm wide, so to give it some room. Variations are possible, see the documentation of adjustbox.

In case a caption is needed, one can use the caption or captionof packages and use something like

\newenvironment{explanation}[2][]
{\noindent\begin{minipage}{\textwidth}\vspace{\topsep}
\begin{minipage}[t]{\dimexpr\textwidth-6cm\relax}}
{\end{minipage}
\par
\if\relax\detokenize\expandafter{\expcaptiontext}\relax\else
\captionof{figure}{\expcaptiontext}
\fi
\gdef\expcaptiontext{}%
\vspace{\topsep}
\end{minipage}}
\newcommand{\expcaption}[1]{\gdef\expcaptiontext{#1}}

with

\begin{explanation}{screenshot1}
This is the text that accompanies the first screen shot
and explains it.

\expcaption{This is the caption}
\end{explanation}

If the text overflows the screen shot by several lines, then the wrapfig package should be taken into consideration.


text:
Detecting Anomalies in a SMART Way (Part Three)

Posted on 20 October 2016

Fortscale's product helps organizations eliminate insider threats by detecting anomalous user behavior.

In the first post of this series I described how we at Fortscale use a personalized adaptive threshold for triggering alerts. Each user’s activity is assigned a risk score (known as the SMART value) that triggers a SMART alert when crosses the user’s threshold. We explained how the more anomalous activities a user performs through time, the higher his threshold gets.

In the second post I dived into technical details, explaining how we use the Bayesian inference framework in order to calculate a score for a SMART value. We developed a formula for the probability of seeing a SMART value of at least $$v$$ for a user with past SMART values $$v_1, ... , v_n$$:

$$\left(\dfrac{\beta_{prior} + \sum_{i=1}^{n} v_i}{\beta_{prior} + \sum_{i=1}^{n} v_i + v}\right)^{\alpha_{prior} + n}$$

Where $$\alpha_{prior}$$ and $$\beta_{prior}$$ are hyper-parameters of the model. In this post we will uncover how we calculate these values.

The prior is a very powerful attribute of the Bayesian inference framework; it allows one to incorporate her prior knowledge when calculating the probability. This attribute is one of the reasons we like Bayesian inference so much. Recall from the first post of the series that we had some challenges with users who never acted anomalously, and hence their past SMART values are all zeros. Such users will surely trigger an alert for any positive SMART value. We can make use of the prior to prevent this from happening.

First, let’s understand how the prior and posterior work together. The prior is the probability function over $$\lambda$$ given no data. Once we observe some data (user generated SMART values), we update our prior and get the posterior, which is again a probability function over $$\lambda$$. Since we chose a conjugate prior (which is the Gamma distribution in the case of the exponential model), we gain the desired property that the update is simply a matter of updating the Gamma distribution’s parameters. If more data is collected, we can update the posterior’s parameters once again. It’s an ongoing process of updating the Gamma distribution’s parameters.

The Gamma distribution’s parameters are $$\alpha$$ and $$\beta$$ (or $$\alpha_{prior}$$ and $$\beta_{prior}$$, since we use the Gamma as a prior). If you’re not familiar with this parameterization, you’re encouraged to read about it here. $$\alpha_{prior}$$ can be interpreted as being the number of SMART values virtually observed by a user, while $$\beta_{prior}$$ is their sum. The user’s real observed data is essentially merged with the virtual data represented by the prior. So if we take the previous example of a user with only zeros as his data, and give a prior with high $$\beta_{prior}$$ to $$\alpha_{prior}$$ ratio, we get the same effect as if the user had some high values, which results in a higher threshold. This is exactly what we wanted to accomplish.

Following the above, we should choose high $$\beta_{prior}$$ to $$\alpha_{prior}$$ ratio. But what should this ratio be? Our goal is to do what an analyst would do when inspecting user activity. The analyst is familiar with the organization and knows which activities could be more interesting, since he has the prior knowledge of how many anomalous activities are going on in the organization. If there’s no anomalous activity at all, any suspicious user activity should be investigated. If anomalous activities is a matter of routine in the organization, the analyst’s interest threshold is higher.

We should simulate this effect using the prior. If we choose $$\alpha_{prior}$$ to be the number of SMART values in the organization and $$\beta_{prior}$$ to be their sum we’ll get the desired effect. Using this method, the prior represents the knowledge of the amount of anomalous activities taking place in the organization.

There is one downside to this approach, namely, it causes $$\alpha_{prior}$$ to be pretty big, which leads to a prior that is too influential. In turn, this will cause that the user’s data will only slightly affect the result probability, i.e. using the posterior will get almost the same results as using the prior. This is not what we aimed for. We wanted the threshold to be personal, which means that the user’s data should affect his threshold.

The reason the prior is too influential is because it simulates SMART values for the user. If there are 10,000 SMART values in the organization $$\alpha_{prior}$$ will be 10,000. If the user had only 100 SMART values, they will be outweighed by the simulated 10,000 SMART values.

If we want to decrease the prior influence, we should decrease $$\alpha_{prior}$$ while preserving the $$\beta_{prior}$$ to $$\alpha_{prior}$$ ratio. The heuristic we chose after experimenting with real life data is to use some reasonable small number, e.g. 20, and to update $$\beta_{prior}$$ to be $$\alpha_{prior}$$ times the average of the organization’s SMART values.

The math behind our choice of the number 20 is simple. The mean of the prior Gamma distribution is $$\alpha_{prior}$$ divided by $$\beta_{prior}$$. The choice of 20 doesn’t affect the mean since we scale $$\beta_{prior}$$ accordingly. The variance however, is $$\alpha_{prior}$$ divided by $$\beta_{prior}$$ squared. Reducing the prior’s “influence” to 20 causes the variance to increase. The intuition behind the prior’s variance is that small variance is analogous to an old stubborn prior that thinks it knows best; when it tells you the expected value is $$\alpha_{prior}$$ divided by $$\beta_{prior}$$, it means exactly that. Big variance on the other hand is like an educated young prior. Such prior knows what the expected value is, but it is not that stubborn and can be “convinced”. Such prior knows there are surprises down the road and consequently allows some uncertainty.

The above is what I meant by the prior’s “influence”: more influence means you should listen to it, and less influence means it gives you some hint, but you should give more weight to the actual user’s data. Choosing the right prior allows us to enjoy both worlds. We get a reasonable prior which takes into account the amount of anomalous activities going on in the organization, while not compromising the desired influence of the user’s data.


text:
# 4x4x4 FMC (computer and human) round 2

#### ch_ts

##### Member
Welcome to 4x4x4 FMC round 2!

Time: May 15 to June 30 2015
April 15 to May 15: Prep time. I thought people might want to tune their solvers using a scramble that will actually count. (see scramble 1 below)

Scoring:

Notation for solutions: SiGN
Please use only the subset of moves that turn a single slice. If your solutions requires moves like Rw or M, please write it using single slices. You may also use rotations x, y, and z. Single slice turn metric will be used for this contest. Any single slice turned any amount will count as 1 move. Rotations count as 0 moves.

Your score will be the sum of your middle three solutions (after dropping the best and worst). In the case of ties, I will consider all 5 solutions, and then I will go by time submitted (earlier is better). You do not need to attempt all 5 scrambles, however, for those you do not attempt, I will score them as 1000 for scoring purposes.

Rules:

You may not use the inverse of the scramble or the inverse of part of the scramble in your solution. You must use the scramble in its entirety; you may not break up the scramble into smaller parts to make use of "mini-scrambles".

Computer FMC: no restrictions, no time limits. You may use code, ideas, programs that you are not the author of. Anything and everything is allowed.
Human FMC: No time limits. Virtual cubes/simulators such as alg.cubing.net are allowed, provided they do not have any solving capabilities. Insertion finder (or similar) is not allowed.

Judge's decisions are final. I may change the rules at any time, I may end the contest early. By participating, you agree to abide by the above rules.

Prizes
Two divisions: computer and human. One prize will be awarded to the winner in the computer division, one prize will be awarded to the winner in the human division.
U.S. winner: 3x3x3 speedcube of your choice (up to $19) or$15 Amazon gift card
International winner: \$15 Amazon gift card

Computer Scrambles:
1. R2 F2 Rw' R D' Uw' L B' L' Rw2 R2 F Uw2 Fw F2 Rw2 B D B2 Rw' F2 Uw2 B' Rw R U Fw' Rw' U' Rw' B' D F2 U Fw' R2 B L2 Uw L2 U D R U2 Uw2
2. B' Fw' L' D F B2 Uw2 D2 R2 L Fw2 F R' Uw' Fw2 B2 Uw Rw2 F L R D U Rw B Fw' R2 Rw2 L' D' B D' F' L2 Uw2 Rw F2 Rw2 F Fw2 U F' Rw' R U
3. Rw F' Rw2 L' Fw2 F2 B2 U' R' L Uw B' R' Rw2 B Fw L2 Rw' Uw2 B' L B2 Fw2 Rw' U Fw Rw2 B R' Uw2 R' B Uw2 B2 Fw2 F2 D' R' B2 U Rw F' R2 Rw2 L
4. B R F' D U' R' B2 D2 B' R B' D2 F' B L2 F' D2 F' U2 R2 Uw2 Rw2 L' B U2 L' B2 Uw2 B' R' D2 B2 Uw Rw2 R2 D2 B2 U' Fw' Rw' B' F2 Uw U' R B2 U' R F2 U
5. F2 L2 B' U2 D2 R2 U2 F U F2 R B R2 D2 B D L' U Fw2 D Rw2 R Fw2 R Uw2 F2 R2 D Rw2 D2 U' Fw F L' Fw2 Rw' Uw F' D2 Rw' B Uw2 B L B U' F2
Submit here

Human Scrambles:
1. R2 F2 Rw' R D' Uw' L B' L' Rw2 R2 F Uw2 Fw F2 Rw2 B D B2 Rw' F2 Uw2 B' Rw R U Fw' Rw' U' Rw' B' D F2 U Fw' R2 B L2 Uw L2 U D R U2 Uw2
2. B2 R Uw Rw2 Uw' Rw' R' Fw U2 Fw2 B U2 Uw B' F L' D' F2 Uw' B Uw2 F' Uw2 U F' D Uw Fw2 B U' F' Uw L' Uw2 U' Fw' F' B' U2 F' Fw L U' B2 Fw2
3. U2 D2 F' D F' Uw2 F L' Rw' U2 F' L Uw2 L' R' Rw' D Uw2 L Uw U' R' Uw2 D B D' L' Rw' Fw' Uw' Rw F' Fw2 Rw2 B2 Rw' U' B' R' Fw Rw F' B U' Uw
4. L' F' U' L2 U' B L D' R' B' R2 L2 D2 F L2 F' L2 U2 R2 F2 Rw2 B' Uw2 F R Fw2 L' Uw2 Rw2 U2 Fw2 F Uw' R2 D B' R' L D2 Fw L2 Fw' Rw' Uw' Rw' R' F R' B2 R2
5. U2 B2 R' U2 R2 B2 U2 F2 R D L' D2 F U2 L' B2 D R' B R2 Fw2 D F2 Rw2 L2 Uw2 D' L' D' L B2 Uw2 R' Fw R2 B' Rw2 U' B' Rw' B Uw' Fw2 B2 L D2 B' U B U D2
Submit here

Last edited:

#### cuBerBruce

##### Member
in my opinion, working on a human solution should not be allowed if the person has previously done any computer searches related to the same scramble. I would strongly recommend not using the same scrambles for both computer and human competitions for this reason.

#### unsolved

##### Member
Welcome to 4x4x4 FMC round 2!
Notation for solutions: SiGN
Just wondering why the scramble is not given in the same notation as the solution requirement.

. In the case of ties, I will consider all 5 solutions, and then I will go by time submitted (earlier is better).
Well yesterday I ordered the hardware for the contest I thought was starting in June, not having seen this announcement. Once it arrives, I have to build the system, test it with the parallel code I am still writing for the 4x4x4, etc. Plus I am also running several 5x5x5 scrambles on my existing hardware for cmowla, Roman, and Stefan, of which you might be presently aware. Just to let everyone know, I will participate, but no doubt be off to a slow start.

#### ch_ts

##### Member
in my opinion, working on a human solution should not be allowed if the person has previously done any computer searches related to the same scramble. I would strongly recommend not using the same scrambles for both computer and human competitions for this reason.
OK, that seems like a good idea. We'll have different scrambles for computers and humans for scramble 2 through 5. I don't want to change the first scramble if people have already started on it.

Just wondering why the scramble is not given in the same notation as the solution requirement.

Well yesterday I ordered the hardware for the contest I thought was starting in June, not having seen this announcement. Once it arrives, I have to build the system, test it with the parallel code I am still writing for the 4x4x4, etc. Plus I am also running several 5x5x5 scrambles on my existing hardware for cmowla, Roman, and Stefan, of which you might be presently aware. Just to let everyone know, I will participate, but no doubt be off to a slow start.
The scramble is the old WCA style scramble but I made a little longer (45 moves OBTM instead of 40). If you're not familiar with WCA scrambles: Uw=u, Rw=r, etc. Most people here are familiar with it. It's a simple extension of SiGN, it's not like a different set of moves like T for top, B for bottom, O for opposite-front. Why did WCA use Rw instead of r? Who knows? That's a discussion for another time. Anyway it shouldn't be too much trouble to scan through it and replace it with moves suitable for your solver.

I'll extend the contest to the end of June.

#### Lucas Garron

##### Super-Duper Moderator
Staff member
Why did WCA use Rw instead of r? Who knows? That's a discussion for another time.
Probably because Ron thought it was more readable. We kept it because of momentum and clarity.
(Think of a competitor using both "u" and "U" in FMC.)

Last edited:

#### cuBerBruce

##### Member
Probably because Ron thought it was more readable. We kept it because of momentum and clarity.
(Think of a competitor using both "u" and "U" in FMC.)
well, I'm pretty sure that the reason had to do with the fact that Singmaster had proposed the lower case letters be used for (single) inner moves on the 4x4x4 back in 1982, and that convention for 4x4x4 was still in widespread use at the time. For example, the popular site bigcubes.com used (and actually still uses) the Singmaster-based notation. I assume the use of lower case letters for wide turns on the 3x3x3 was started much later (than 1982). The "w" notation apparently eventually had become widespread in the Asian community and thus seemed to provide an unambiguous notation for the WCA to use, rather using a notation that people might be confused about what it meant.

#### ch_ts

##### Member
Just posted another scramble. Make sure you're using the correct scrambles, as we'll be using different scrambles for the computer and human contests following cuBerBruce's suggestion. Just another half month before we're officially under way!

#### unsolved

##### Member
Just posted another scramble. Make sure you're using the correct scrambles, as we'll be using different scrambles for the computer and human contests following cuBerBruce's suggestion. Just another half month before we're officially under way!
I have almost debugged an awesome new pruning mechanism that treats forward move generation the same way my TFS databases treats the back-end of the search

All edges + centers pruning tables have been solved through 6 turns, and corners through 8 turns. I have enough RAM for this + TFS_05. Still needs some implementation debugging.

Look at these speeds...

Computer Scrambles:
1. R2 F2 Rw' R D' Uw' L B' L' Rw2 R2 F Uw2 Fw F2 Rw2 B D B2 Rw' F2 Uw2 B' Rw R U Fw' Rw' U' Rw' B' D F2 U Fw' R2 B L2 Uw L2 U D R U2 Uw2
So that would be this position, correct?

https://alg.cubing.net/?puzzle=4x4x..._Rw-_B-_D_F2_U_Fw-_R2_B_L2_Uw_L2_U_D_R_U2_Uw2

Last edited by a moderator:

#### ch_ts

##### Member
OK, I've posted the scrambles and we have officially started. Good luck and happy cubing!

#### rokicki

##### Member
Woo hoo!

OK, I've posted the scrambles and we have officially started. Good luck and happy cubing!
Wow! Good luck all!

Is there any automatic validation of our solutions, or are you doing that manually?

It might be nice to know if the solutions worked or if we made a silly error.

#### ch_ts

##### Member
Yes, I'll be doing that manually by pasting them into alg.cubing.net. I never got around to writing a script to handle the entries. Anyway, I don't expect there to be so many entries that I'll be overwhelmed by them (judging by the number of entries in last year's contest)

#### unsolved

##### Member
Maybe the programmers can announce their presence, the names of their programs, and any details about their solvers that won't jeapordize their secret sauce

Program Name = Omnia Obtorquebantur 4x4x4 version 3.0.8

Running on: Intel i7-3960X @ 4.7 GHz with 32 GB RAM.

Features:

1. Forward Pruning Table #1 = Corners + Edges to depth 5
2. Forward Pruning Table #2 = Centers to depth 6
3. Back-End Pruning Table #3 = Entire Cube to depth 5
4. Fast 64-bit, Facelet-Level Move Generator, only 6 lines of C-code executed to generate any move

Pros:

1. Deep pruning capabilities
2. Generates only theoretical minimum tree size at each depth (before pruning taken into account)
3. 2D graphical representation of the scramble
4. Can solve in-line rotations along with the scramble
5. Text output file lists every solution found
6. New stage solving mode (though highly experimental)

Cons:

1. Still needs 24 different rotated versions of all pruning data, bogging down memory
2. Spends most of its time exploring 0 nodes, iterating in for-loops in search of theoretical minimal move point of entrance to solution(s).
3. Does not yet fully support SiGN in move parsing and solution output (but 5x5x5 version does)

Last edited:

#### qq280833822

##### Member
Program Name = Two Phase Reduction Solver + min2phase with huge table (instead of cube explorer).

Running on: Intel core i7-2670QM @2.2 GHz with 8 GB RAM.

All 5 scrambles have been solved no more than 45 moves, wherein the reduction parts of these scrambles are no more than 27 moves. (all in SSTM)

==========update==========
Huge table is equivalent to the 1.8G table used in cube explorer to accelerate the search of 3x3x3.

Last edited:

#### StachuK1992

##### statue
The reduction parts were no more than 27moves, having an upper limit of 27 + 20 = 47 moves total (plus some for parities?).
That said it seems the ones with 27moves had 18 move 3x3 solutions

#### unsolved

##### Member
The reduction parts were no more than 27moves, having an upper limit of 27 + 20 = 47 moves total (plus some for parities?).
That said it seems the ones with 27moves had 18 move 3x3 solutions
I have to disqualify my 36-move result after chatting with one of the other participants. I have an algorithm that makes use of subsets of the scramble. I started working on it long before the contest when I was making my own stage solver, and I discovered an interesting way to find near-optimal or optimal solutions to just about any scramble. It's very CPU-intensive and massively parallel. So, until I come up with a new stage solving algorithm, I must remove myself.

#### qq280833822

##### Member
I have to disqualify my 36-move result after chatting with one of the other participants. I have an algorithm that makes use of subsets of the scramble. I started working on it long before the contest when I was making my own stage solver, and I discovered an interesting way to find near-optimal or optimal solutions to just about any scramble. It's very CPU-intensive and massively parallel. So, until I come up with a new stage solving algorithm, I must remove myself.
You mean you have solved one of these scrambles in 36 moves, or just finished the reduction part in 36 moves?

#### Herbert Kociemba

##### Member
I have to disqualify my 36-move result after chatting with one of the other participants. I have an algorithm that makes use of subsets of the scramble. I started working on it long before the contest when I was making my own stage solver, and I discovered an interesting way to find near-optimal or optimal solutions to just about any scramble. It's very CPU-intensive and massively parallel. So, until I come up with a new stage solving algorithm, I must remove myself.
I do not think you have to come with a new stage algorithm. If you really have found an algorithm which finds near-optimal or optimal(!!!) solutions to just about any scramble this would be a phantastic piece of work and of course these solutions would be accepted by all other participants. SUBMIT THESE SOLUTIONS! If not I suppose these solutions are so phantastic that they only exist in your fantasies and you have a creative handling of truth.

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#### qqwref

##### Member
My algorithm was nothing more than a fancy line-crawl @ fixed depth, solving subsets of the scramble, and rearranging the solution at the end. It reduced the solution length from 52 moves in my program's metric to 36 moves, a savings of 16 moves. The very first line of the rules forbids this very thing. I never read it. I just processed the scrambles.
Disregarding the point that this isn't even a valid FMC technique, how are you sure this will still solve the scramble? Because of 4x4x4's sets of identical centers, it's entirely possible to have move sequences A, B, C such that B and C do the same thing to a solved cube, but A B and A C don't.

Even though I only have distances of 5- and 6- respectively, they are partially additive in nature (depending on the position) and there is no more effective way to forward prune, I guarantee it.
What does "partially additive" even mean? And the last part of this is rather silly.

I don't use an indexing function, I use a hash table, which more than one of the posters on here treated with derision.
Probably because you act like it's some kind of impressive and extremely clever technique, rather than one of the fundamental data structures that even a beginning programmer ought to be familiar with.


text:
International Journal of Mathematics ( IF 0.604 ) Pub Date : 2020-09-19 , DOI: 10.1142/s0129167x20500925
Myungho Kim; Euiyong Park

In this paper, we investigate a connection between convolution products for quiver Hecke algebras and tensor products for quantum groups. We give a categorification of the natural projection $πλ,μ:V𝔸(λ)∨⊗𝔸V𝔸(μ)∨↠V𝔸(λ+μ)∨$ sending the tensor product of the highest weight vectors to the highest weight vector in terms of convolution products. When the quiver Hecke algebra is symmetric and the base field is of characteristic $0$, we obtain a positivity condition on some coefficients associated with the projection $πλ,μ$ and the upper global basis, and prove several results related to the crystal bases. We then apply our results to finite type $A$ using the homogeneous simple modules $𝒮T$ indexed by one-column tableaux $T$.

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