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# zbMATH — the first resource for mathematics

## Glasgow Mathematical Journal

 Short Title: Glasg. Math. J. Publisher: Cambridge University Press, Cambridge ISSN: 0017-0895; 1469-509X/e Online: https://www.cambridge.org/core/journals/glasgow-mathematical-journal/all-issues Predecessor: Proceedings of the Glasgow Mathematical Association Comments: Indexed cover-to-cover
 Documents Indexed: 2,305 Publications (since 1967) References Indexed: 1,821 Publications with 21,818 References.
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#### Latest Issues

 63, No. 3 (2021) 63, No. 2 (2021) 63, No. 1 (2021) 62, No. 3 (2020) 62, No. 2 (2020) 62, No. S1 (2020) 62, No. 1 (2020) 61, No. 3 (2019) 61, No. 2 (2019) 61, No. 1 (2019) 60, No. 3 (2018) 60, No. 2 (2018) 60, No. 1 (2018) 59, No. 3 (2017) 59, No. 2 (2017) 59, No. 1 (2017) 58, No. 3 (2016) 58, No. 2 (2016) 58, No. 1 (2016) 57, No. 3 (2015) 57, No. 2 (2015) 57, No. 1 (2015) 56, No. 3 (2014) 56, No. 2 (2014) 56, No. 1 (2014) 55A (2013) 55, No. 3 (2013) 55, No. 2 (2013) 55, No. 1 (2013) 54, No. 3 (2012) 54, No. 2 (2012) 54, No. 1 (2012) 53, No. 3 (2011) 53, No. 2 (2011) 53, No. 1 (2011) 52A (2010) 52, No. 3 (2010) 52, No. 2 (2010) 52, No. 1 (2010) 51A (2009) 51, No. 3 (2009) 51, No. 2 (2009) 51, No. 1 (2009) 50, No. 3 (2008) 50, No. 2 (2008) 50, No. 1 (2008) 49, No. 3 (2007) 49, No. 2 (2007) 49, No. 1 (2007) 48, No. 3 (2006) 48, No. 2 (2006) 48, No. 1 (2006) 47A (2005) 47, No. 3 (2005) 47, No. 2 (2005) 47, No. 1 (2005) 46, No. 3 (2004) 46, No. 2 (2004) 46, No. 1 (2004) 45, No. 3 (2003) 45, No. 2 (2003) 45, No. 1 (2003) 44, No. 3 (2002) 44, No. 2 (2002) 44, No. 1 (2002) 43A (2001) 43, No. 3 (2001) 43, No. 2 (2001) 43, No. 1 (2001) 42, No. 3 (2000) 42, No. 2 (2000) 42, No. 1 (2000) 41, No. 3 (1999) 41, No. 2 (1999) 41, No. 1 (1999) 40, No. 3 (1998) 40, No. 2 (1998) 40, No. 1 (1998) 39, No. 3 (1997) 39, No. 2 (1997) 39, No. 1 (1997) 38, No. 3 (1996) 38, No. 2 (1996) 38, No. 1 (1996) 37, No. 3 (1995) 37, No. 2 (1995) 37, No. 1 (1995) 36, No. 3 (1994) 36, No. 2 (1994) 36, No. 1 (1994) 35, No. 3 (1993) 35, No. 2 (1993) 35, No. 1 (1993) 34, No. 3 (1992) 34, No. 2 (1992) 34, No. 1 (1992) 33, No. 3 (1991) 33, No. 2 (1991) 33, No. 1 (1991) 32, No. 3 (1990) ...and 29 more Volumes
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#### Authors

 15 Petrich, Mario 11 Chō, Muneo 9 Crabb, Michael J. 9 Luca, Florian 9 O’Regan, Donal 9 Shparlinski, Igor E. 9 Stothers, W. Wilson 8 Duggal, Bhagwati Prashad 8 May, Coy L. 7 Chatters, A. W. 7 Cohen, Stephen D. 7 Jordan, David Alan 7 López-Permouth, Sergio Roberto 7 McGregor, Colin M. 7 Nicholson, W. Keith 7 Smith, Howard 6 Anderson, Ian 6 Berndt, Bruce Carl 6 Chen, Bang-Yen 6 de Giovanni, Francesco 6 Duncan, John 6 Gromadzki, Grzegorz 6 Humphries, Stephen P. 6 Huynh, Dinh Van 6 Jeon, In Ho 6 Ko, Eungil 6 Puczyłowski, Edmund R. 6 Singerman, David 6 Smith, Patrick Fleming 6 Yousif, Mohamed F. 6 Zhou, Yiqiang 5 Assem, Ibrahim 5 Ballester-Bolinches, Adolfo 5 Beazer, Rod 5 Birkenmeier, Gary Francis 5 Cossey, John 5 Franciosi, Silvana 5 González Ortiz, Manuel 5 Gould, Victoria A. R. 5 Griggs, Terry S. 5 Holub, James R. 5 Jamison, James Edward 5 Jones, Peter R. 5 Kappe, Luise-Charlotte 5 Kittaneh, Fuad 5 Lamont, Patrick J. C. 5 Lange, Ridgley 5 Lenagan, Thomas H. 5 Lowndes, John S. 5 Lucchini, Andrea 5 McAlister, Donald B. 5 Odoni, R. W. K. 5 Reilly, Norman R. 5 Sharp, Rodney Y. 5 Shumyatsky, Pavel 5 Tweddle, Ian 5 Van Maldeghem, Hendrik J. 5 Wehrfritz, Bertram A. F. 5 Zaharescu, Alexandru 5 Zhang, James J. 4 Agarwal, Ravi P. 4 Akemann, Charles A. 4 Amini, Massoud 4 Au-Yeung, Yik-Hoi 4 Azizi, Abdulrasool 4 Bavula, Vladimir V. 4 Beidleman, James Clark 4 Blyth, Tom S. 4 Bonsall, Frank Featherstone 4 Brandl, Rolf 4 Brown, Kenneth Alexander 4 Brüdern, Jörg 4 Burns, John M. 4 Chiappinelli, Raffaele 4 Costa Gonzáles, Antonio Félix 4 Ellis, Graham J. 4 Harman, Glyn 4 Hidalgo, Rubén Antonio 4 Higgins, Peter Michael 4 Hooley, Christopher 4 Iyahen, S. O. 4 Jespers, Eric 4 Kalton, Nigel John 4 Koubek, Vaclav 4 Lambert, Alan 4 Leavitt, William G. 4 Liu, Zhong-kui 4 Longobardi, Patrizia 4 Maclachlan, Colin 4 Magill, Kenneth D. jun. 4 Maj, Mercede 4 McFadden, Robert B. 4 Mignotte, Maurice 4 Mihăilescu, Mihai 4 Munn, Walter Douglas 4 Neeb, Karl-Hermann 4 Nimmo, Jonathan J. C. 4 Park, Jae Keol 4 Robinson, Paul Lee 4 Saab, Paulette ...and 2,403 more Authors
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#### Fields

 464 Group theory and generalizations (20-XX) 308 Associative rings and algebras (16-XX) 287 Functional analysis (46-XX) 274 Operator theory (47-XX) 262 Number theory (11-XX) 134 Differential geometry (53-XX) 120 Partial differential equations (35-XX) 98 Functions of a complex variable (30-XX) 97 Manifolds and cell complexes (57-XX) 96 Algebraic geometry (14-XX) 95 Combinatorics (05-XX) 93 Commutative algebra (13-XX) 83 Nonassociative rings and algebras (17-XX) 70 Dynamical systems and ergodic theory (37-XX) 63 Category theory; homological algebra (18-XX) 62 Order, lattices, ordered algebraic structures (06-XX) 59 Ordinary differential equations (34-XX) 59 Algebraic topology (55-XX) 47 Global analysis, analysis on manifolds (58-XX) 44 Several complex variables and analytic spaces (32-XX) 38 Linear and multilinear algebra; matrix theory (15-XX) 38 Topological groups, Lie groups (22-XX) 37 General topology (54-XX) 28 General algebraic systems (08-XX) 28 Real functions (26-XX) 28 Special functions (33-XX) 26 Harmonic analysis on Euclidean spaces (42-XX) 25 Field theory and polynomials (12-XX) 25 Integral equations (45-XX) 23 Measure and integration (28-XX) 23 Abstract harmonic analysis (43-XX) 22 Probability theory and stochastic processes (60-XX) 21 Integral transforms, operational calculus (44-XX) 19 Convex and discrete geometry (52-XX) 17 Difference and functional equations (39-XX) 17 Geometry (51-XX) 14 Quantum theory (81-XX) 11 Potential theory (31-XX) 11 Fluid mechanics (76-XX) 10 History and biography (01-XX) 10 $$K$$-theory (19-XX) 10 Sequences, series, summability (40-XX) 10 Calculus of variations and optimal control; optimization (49-XX) 10 Computer science (68-XX) 9 Mathematical logic and foundations (03-XX) 9 Approximations and expansions (41-XX) 6 Numerical analysis (65-XX) 5 Mechanics of deformable solids (74-XX) 5 Statistical mechanics, structure of matter (82-XX) 5 Information and communication theory, circuits (94-XX) 4 General and overarching topics; collections (00-XX) 4 Systems theory; control (93-XX) 3 Mechanics of particles and systems (70-XX) 3 Classical thermodynamics, heat transfer (80-XX) 2 Statistics (62-XX) 2 Relativity and gravitational theory (83-XX) 2 Operations research, mathematical programming (90-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX)

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Asymptotic behavior of eigenvalues of reciprocal power LCM matrices. Zbl 1186.11011
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Solving the truncated moment problem solves the full moment problem. Zbl 0995.44004
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A bilinear approach to a Pfaffian self-dual Yang-Mills equation. Zbl 0986.35109
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2001
Hopf images and inner faithful representations. Zbl 1243.16033
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2010
Poor modules: the opposite of injectivity. Zbl 1228.16004
2010
Valuations of near polygons. Zbl 1085.51009
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Axiomatisations of the average and a further generalisation of monotonic sequences. Zbl 0291.40003
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On Littlewood-Paley functions associated with Bessel operators. Zbl 1151.42008
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Star-complexes, and the dependence problems for hyperbolic complexes. Zbl 0684.20022
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1988
On non-Hurwitz groups and non-congruence subgroups of the modular group. Zbl 0452.20049
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1981
Extended eigenvalues and the Volterra operator. Zbl 1037.47013
Biswas, Animikh; Lambert, Alan; Petrovic, Srdjan
2002
Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip. Zbl 1142.35067
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2007
Stability of mappings on multi-normed spaces. Zbl 1125.39023
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2007
A spectral mapping theorem for the Weyl spectrum. Zbl 0869.47017
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1996
Consistency conditions for dimer models. Zbl 1244.14042
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The lightcone Gauss map and the lightcone developable of a spacelike curve in Minkowski 3-space. Zbl 0970.53015
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2000
Non-abelian exterior products of groups and exact sequences in the homology of groups. Zbl 0631.20040
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1991
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1978
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2006
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Maps preserving the local spectrum of product of operators. Zbl 1326.47041
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A note on inhomogeneous Diophantine approximation. Zbl 1039.11048
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Biembeddings of Latin squares and Hamiltonian decompositions. Zbl 1062.05030
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2004
On the family of Diophantine triples $$\{k-1, k+1, 16k^3 - 4k\}$$. Zbl 1168.11007
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2010
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2005
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1987
Boundaries of nonpositively curved groups of the form $$G\times \mathbb{Z}^ n$$. Zbl 0861.53044
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1996
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1978
On Weyl and Browder spectra of tensor products. Zbl 1136.47013
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2008
A simple proof of the Goldberg-Straus theorem on numerical radii. Zbl 0605.15019
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1986
Kernels of representations and coideal subalgebras of Hopf algebras. Zbl 1245.16024
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2012
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2009
Hermite function expansions versus Hermite polynomial expansions. Zbl 1129.42409
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2006
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1968
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1992
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1999
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1997
On commutative Noetherian rings which satisfy the radical formula. Zbl 0921.13001
Leung, Ka Hin; Man, Shing Hing
1997
Detecting Steiner and linear isometries operads. Zbl 1464.55018
Rubin, Jonathan
2021
Closed ideals and Lie ideals of minimal tensor product of certain $$C$$*-algebras. Zbl 07333635
Talwar, Bharat; Jain, Ranjana
2021
On nonlocal nonlinear elliptic problems with the fractional Laplacian. Zbl 1429.35092
Ma, Li
2020
Flat ring epimorphisms of countable type. Zbl 1445.16002
Positselski, Leonid
2020
Local negativity of surfaces with non-negative Kodaira dimension and transversal configurations of curves. Zbl 1445.14016
Laface, Roberto; Pokora, Piotr
2020
Oriented flip graphs, noncrossing tree partitions, and representation theory of tiling algebras. Zbl 07141357
Garver, Alexander; McConville, Thomas
2020
The unital Ext-groups and classification of $$C^*$$-algebras. Zbl 1440.46051
Gabe, James; Ruiz, Efren
2020
Epsilon-strongly groupoid-graded rings, the Picard inverse category and cohomology. Zbl 07141361
Nystedt, Patrik; Öinert, Johan; Pinedo, Héctor
2020
Class number formula for dihedral extensions. Zbl 1450.11117
Caputo, Luca; Nuccio Mortarino Majno Di Capriglio, Filippo A. E.
2020
Decomposing the tube category. Zbl 1448.18033
Hardiman, Leonard; King, Alastair
2020
Nakayama automorphisms of Ore extensions over polynomial algebras. Zbl 1455.16010
Liu, Liyu; Ma, Wen
2020
Configuration categories and homotopy automorphisms. Zbl 1436.57036
Weiss, Michael S.
2020
Characterizations of Berger spheres from the viewpoint of submanifold theory. Zbl 1430.53064
Kim, Byung Hak; Kim, In-Bae; Maeda, Sadahiro
2020
How small can polynomials be in an interval of given length? Zbl 1439.11175
Bernik, Vasili; Mc Guire, Stephen
2020
A note on $$A$$-annihilated generators of $$H_* QX$$. Zbl 1442.55013
2020
On the arithmetic of Mori monoids and domains. Zbl 1440.13005
Zhong, Qinghai
2020
JSJ decompositions of doubles of free groups. Zbl 07194255
Heil, Simon
2020
Localizations of the hearts of cotorsion pairs. Zbl 07236510
Liu, Yu
2020
Fourier-type transforms on rearrangement-invariant quasi-Banach function spaces. Zbl 1440.42018
Ho, Kwok-Pun
2019
Exhaustion of the curve graph via rigid expansions. Zbl 06988392
Hernández Hernández, Jesús
2019
Distortion in the finite determination result for embeddings of locally finite metric spaces into Banach spaces. Zbl 1459.46028
Ostrovska, S.; Ostrovskii, M. I.
2019
Some results of the $$\mathcal{K}_{\mathcal{A}}$$-approximation property for Banach spaces. Zbl 1430.46013
Kim, Ju Myung
2019
Frequently hypercyclic bilateral shifts. Zbl 07114290
Grosse-Erdmann, Karl-G.
2019
Homotopy types of gauge groups over non-simplyconnected closed 4-manifolds. Zbl 1429.55008
So, Tseleung
2019
Focal surfaces of wave fronts in the Euclidean 3-space. Zbl 1429.57028
Teramoto, Keisuke
2019
Complex of relatively hyperbolic groups. Zbl 07093785
Pal, Abhijit; Paul, Suman
2019
Frame-less Hilbert C*-modules. Zbl 1458.46046
Asadi, M. B.; Frank, M.; Hassanpour-Yakhdani, Z.
2019
Prime spectra of ambiskew polynomial rings. Zbl 1445.16024
Fish, Christopher D.; Jordan, David A.
2019
Complex product structures on Hom-Lie algebras. Zbl 06988387
2019
On degenerate sigma-functions in genus 2. Zbl 1439.14097
Bernatska, Julia; Leykin, Dmitry
2019
Higgs bundles over elliptic curves for complex reductive groups. Zbl 1433.14028
2019
Strong approximation theorem for absolutely irreducible varieties over the compositum of all symmetric extensions of a global field. Zbl 1439.12008
Jarden, Moshe; Razon, Aharon
2019
A structured description of the genus spectrum of abelian $$p$$-groups. Zbl 07114296
Müller, Jürgen; Sarkar, Siddhartha
2019
A perturbation and generic smoothness of the Vafa-Witten moduli spaces on closed symplectic four-manifolds. Zbl 1423.53102
Tanaka, Yuuji
2019
Extreme points for combinatorial Banach spaces. Zbl 1435.46009
Beanland, Kevin; Duncan, Noah; Holt, Michael; Quigley, James
2019
Stratifying systems for exact categories. Zbl 1441.18011
Santiago, Valente
2019
Test map and discreteness in $$\mathrm{SL}(2,\mathbb{H})$$. Zbl 07093776
Gongopadhyay, Krishnendu; Mukherjee, Abhishek; Sardar, Sujit Kumar
2019
The curved $$A_\infty$$-coalgebra of the Koszul codual of a filtered dg algebra. Zbl 07093780
Herscovich, Estanislao
2019
$$r$$-Fredholm theory in Banach algebras. Zbl 1431.46030
Benjamin, Ronalda; Laustsen, Niels Jakob; Mouton, Sonja
2019
On the axiomatizability of $$C^*$$-algebras as operator systems. Zbl 1431.46036
Goldbring, Isaac; Sinclair, Thomas
2019
Nonincreasing depth functions of monomial ideals. Zbl 1421.13001
Matsuda, Kazunori; Suzuki, Tao; Tsuchiya, Akiyoshi
2018
Braided mixed datums and their applications on Hom-quantum groups. Zbl 1443.18010
Zhang, Xiaohui; Dong, Lihong
2018
Pure-injectivity from a different perspective. Zbl 1441.16004
López-Permouth, S. R.; Mastromatteo, J.; Tolooei, Y.; Ungor, B.
2018
Koszul calculus. Zbl 1431.16027
Berger, Roland; Lambre, Thierry; Solotar, Andrea
2018
A moduli interpretation for the non-split Cartan modular curve. Zbl 1440.11106
Rebolledo, Marusia; Wuthrich, Christian
2018
Nonnoetherian homotopy dimer algebras and noncommutative crepant resolutions. Zbl 1453.16013
Beil, Charlie
2018
Non-negative deformations of weighted homogeneous singularities. Zbl 1380.32028
Nuño-Ballesteros, J. J.; Oréfice-Okamoto, B.; Tomazella, J. N.
2018
Schwarz lemma for holomorphic mappings in the unit ball. Zbl 1384.32005
Kalaj, David
2018
Characterizations of locally finite actions of groups on sets. Zbl 1387.43002
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text:
# Thread: quick help with the derivative of e^Wt

1. ## quick help with the derivative of e^Wt

Hi Guys,
Not posted on here before. Just found this place!
The question is : whats the derivative of e to the power of omega T.
Im stuck as im not sure as the omega and the T are variables so does the omega get put in front of the e?
Am i making any sense?

Thanks, James.

2. Originally Posted by james jarvis
Hi Guys,
Not posted on here before. Just found this place!
The question is : whats the derivative of e to the power of omega T.
Im stuck as im not sure as the omega and the T are variables so does the omega get put in front of the e?
Am i making any sense?

Thanks, James.
just the T is a variable. the omega is a constant, so treat it as such. you'd do the same thing as if you had e^(2t) or something

3. Originally Posted by james jarvis
Hi Guys,
Not posted on here before. Just found this place!
The question is : whats the derivative of e to the power of omega T.
Im stuck as im not sure as the omega and the T are variables so does the omega get put in front of the e?
Am i making any sense?

Thanks, James.

$\displaystyle e^{T\omega} \frac{\mathrm{d}}{\mathrm{dT}}$$\displaystyle = \omega e^{T\omega}$

4. Thanks,
Some how convinced myself that they were both variables.
James.

,

,

,

,

,

,

# differentiation of x not e to the power i(wt-kna)

Click on a term to search for related topics.

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text:
# Henselianity in the language of rings, with Franziska Jahnke

I would like to write about a new paper ([AJ15]) which Franziska Jahnke and I have written and put on the arXiv. It’s called Henselianity in the language of rings. In it we investigate the relationship between the following four properties of a field $K$:

(h) $K$ is henselian, i.e. $K$ admits a nontrivial henselian valuation,

(eh) $K$ is elementarily henselian, i.e. every $L\equiv K$ is henselian,

(def) $K$ admits a definable nontrivial henselian valuation, and

($\emptyset$-def) $K$ admits a $\emptyset$-definable nontrivial henselian valuation.

By definable’ we mean definable in the language of rings’. Our paper is the latest work on this topic, which began with the work of Prestel and Ziegler who show, in [PZ78], that there exist t-henselian, nonhenselian fields. Subsequently, several more recent papers explore this topic and related issues of definable valuations; examples include [JK15] and [FJ15]. Our work builds on these three papers, and others. A more complete survey of the literature is available in our paper.

There are some trivial implications between the above properties:

${\bf(\emptyset\textbf{-def})}\Longrightarrow{\bf(\textbf{def})}\Longrightarrow{\bf(h)}\qquad\qquad(a)$

and

${\bf(\emptyset\textbf{-def})}\Longrightarrow{\bf(eh)}\Longrightarrow{\bf(h)}.\qquad\qquad(b)$

Franziska and I investigate whether any other implications hold between these properties across the class $\mathcal{K}_{0}$ of all fields of characteristic zero, and across the smaller classes

$\mathcal{K}_{0,0}:=\{K\in\mathcal{K}_{0}\;|\;\mathrm{char}(Kv_{K})=0\}$

and

$\mathcal{K}_{0,p}:=\{K\in\mathcal{K}_{0}\;|\;\mathrm{char}(Kv_{K})=p\}$, for $p$ a prime;

where $v_{K}$ denotes the canonical henselian valuation on $K$. Our main theorem gives the complete picture’ for each of these classes $\mathcal{K}_{0,0},\mathcal{K}_{0,p}:$

Main Theorem (Theorem 1.1, [AJ15])

1. In the class ${\mathcal{K}_{0,0}}$ the complete picture is

${\bf(eh)} \Longleftrightarrow {\bf(\emptyset\textbf{-def})} \Longrightarrow {\bf(\textbf{def})} \Longrightarrow {\bf(h)}.$

2. For each prime ${p}$, in the class ${\mathcal{K}_{0,p}}$ the complete picture is

${\bf(\textbf{def})} \Longleftrightarrow {\bf(\emptyset\textbf{-def})} \Longrightarrow {\bf(eh)} \Longleftrightarrow {\bf(h)}.$

3. Consequently, in the class ${\mathcal{K}_{0}}$ the complete picture is simply given by (1) and (2), above.

Given the trivial’ implications, (a) and (b) above, the proof of Theorem 1.1 boils down to showing the following:

1. In the class $\mathcal{K}_{0,0}$ the implication ${\bf(eh)}\Longrightarrow{\bf(\emptyset\textbf{-def})}$ holds.
2. For each prime $p$, in the class $\mathcal{K}_{0,p}$ the implications ${\bf(h)}\Longrightarrow{\bf(eh)}$ and ${\bf(def)}\Longrightarrow{\bf(\emptyset\textbf{-def})}$ hold.
3. No implications hold other than the ones listed above and the trivial’ ones.

The proof of (4) has two stages: first we show that ${\bf(eh)}\Longrightarrow{\bf(def)}$ using Theorem B from [JK15]; then we employ the Omitting Types Theorem to remove the parameter.

The proof of (5) is more tricky. The implication ${\bf(h)}\Longrightarrow{\bf(eh)}$ follows from basic properties of the canonical henselian valuation combined with some model-theoretic trickery. Franziska and I call these arguments Sirince tricks’ because we wrote them down when we were on a research visit to the wonderful Nesin Mathematics Village, near Sirince, Turkey! The proof of the implication ${\bf(def)}\Longrightarrow{\bf(\emptyset\textbf{-def})}$ depends on some delicate arguments involving uniform definitions of canonical $p$-henselian valuations.

Finally, the proof of (6) is a set of four counterexamples. The first (given by Prestel-Ziegler, [PZ78]) simply shows the existence (in equicharacteristic zero) of t-henselian nonhenselian fields. The second and third examples were given in [JK15] and are based on the construction in [PZ78]. Finally, the fourth example is based on a more sophisticated construction give in [FJ15] of t-henselian nonhenselian fields with various properties – we extend it to defectless fields of positive characteristic.

There are several key open questions. In my opinion, perhaps the most prominent is: can our results be extended to equicharacteristic $p>0$?

#### Bibliography

[AJ15] Sylvy Anscombe and Franziska Jahnke. Henselianity in the language of rings. Manuscript, 2015. arXiv
[FJ15] Arno Fehm and Franziska Jahnke. On the quantifier complexity of definable canonical henselian valuations. Mathematical Logic Quarterly, 61(4-5):347-361, 2015.
[JK15] Franziska Jahnke and Jochen Koenigsmann. Defining coarsenings of valuations. To appear in the Proceedings of the Edinburgh Mathematical Society, 2015. arXiv
[PZ78] Alexander Prestel and Martin Ziegler. Model-theoretic methods in the theory of topological fields. Journal für die reine und angewandte Mathematik, 299(300):318-341, 1978.


text:
# The Machinery of Morality

Part Of: Demystifying Ethics sequence
Content Summary: 1000 words, 10 min read

Propriety Frames

Main Article

The primate brain contains a diverse set of memory structures. For example, episodic and semantic memory store narratives and facts, respectively.

Propriety frames are a memory format that retains social intuitions. This form of memory permits normative judgments, or which behaviors are appropriate or inappropriate.

Propriety frames are organized by situational context. A Restaurant frame provides the social intuitions for restaurant interactions. Frames are the substrate of rituals.

Frames are organized hierarchically to promote reuse. For example, the Eating frame is relevant in contexts besides restaurant dining.

When a mother instructs her son to not yell in the store, the child installs an update to his Shopping frame. When a family exchanges gossip around a campfire, they are doing so in part to synchronize their propriety frames.

Intuition Generators

The contents of our social intuitions is not arbitrary. Our environment does not fully determine our moral sense. The brain also possesses innate social intuition generators, which contribute to the contents of social judgments.

As outlined in Moral Foundations theory, there are six such generators: authority, care, loyalty, fairness, autonomy, and sanctity.

People are genetically and environmentally disposed to respond to certain generators more strongly than others. Let social matrices be the emotional intensities elicited by each generator.

People with similar matrices tend to gravitate towards similar political parties. When you measure the social matrices of American citizens, you can see large differences between the social intuitions of Democrats and Republicans.

These differences in social matrices explain much of American politics.

• Why do Democrats praise entitlements, but Republicans denounce them? Because Democrats heavily emphasize Care for the poor, whereas Republicans more strongly reverberate to questions of Fairness (exploitation and moral hazard).
• Why are Democrats more skeptical of patriotism than their Republican counterparts? Perhaps because they respond to Loyalty to country less.

For more information, see Graham et al (2009). Liberals and conservatives rely on different sets of moral foundations.

Generators vs Frames

Main Article: A Dual-Process Theory of Moral Judgment

Two mechanisms contribute to social intuitions: propriety frames, and intuition generators.

Generators are located subcortically, and thus produce intuitions quickly and automatically. Frames are stored in the cortical mantle, and are thus slower, and more amenable to conscious awareness.

As the work of Joshua Greene shows, deontological (rule-based) attitudes are generated rapidly; whereas consequential theories of morality are slower and more vulnerable to distraction. This suggests a relatively straightforward mapping:

• People with a deontological style rely more heavily on their Generators.
• People with a consequentialist style lean instead on their Frames.

Frames and Generators can influence one another, albeit slowly. Moral argumentation can change one’s mind, and these frame updates can percolate down to change one’s social matrices (intuition weights). Call this frame internalization.

Similarly, if one’s private intuitions diverge from a culturally inherited norm, that frame can be updated to be more consistent with one’s personality. Call this frame refinement.

For more information, see Cushman, Young, Greene (2010). Our multi-system moral psychology: Towards a consensus view

Cultural Regimes

Human communities generate emergent networks known as cultural regimes. These emerge as distinct categories of frames and intuition generators.

Language made regime accretion possible. No longer is expertise confined to the skull of the individual. No longer is death a Full Restart button. Our species sends cultural expertise down through the generations.  This knowledge – these frames – have become increasingly sophisticated over the course of human history.

Hominid evolution has also seen the advent of regime diversification. Modern religions (of the Axial Age) were successful because they improved on our ability to trust strangers. This in turn dramatically increased the size of feasible social groups.

Moral Tagging

We have so far only been speaking about social attitudes (should / should not). What about moral attitudes (good / evil)?

Moral attitudes are nothing more than a kind of social attitude. I know of no moral attitude that can be divorced from a social context.

Moral attitudes are constructed by moral tagging, which endows a subset of social attitudes with anger and disgust reactions (as opposed to the more typical reputation appraisal, gossiping, shaming).

Further, moral tagging produces appraisal inflexibility. Moral violations are viewed as wrong everywhere, in every context. This is in contrast to other social violations, for which it is easier to see circumstances in which the behavior would seem less bad.

The boundary between Virtuous and Tolerable is smooth, reflecting the flexibility of our intuition generators. In contrast, the boundary between Tolerable and Intolerable is sharp.

A similar distinction appears amidst disagreement. There are two kinds of ways people’s judgments can diverge:

• Moral disagreement: is a given behavior Evil?
• Social disagreement: is a given behavior Inappropriate?

For more information, see Tetlock et al (2000). The psychology of the unthinkable: taboo trade-offs, forbidden base rates, and heretical counterfactuals.

Norm Synchronization

The space of possible social intuitions is vast. However, group living only became possible with relatively homogenous norms. How do individual brains synchronize propriety frames and social matrices within a group?

At least four mechanisms provide norm synchronization.

• Natural language and facial expressions are used to communicate social and emotional information.
• Reputation systems benefit social beings who are especially adept at learning & conforming to the frames of their peers.
• First-order punishment emotions (anger at transgressor) incentivizes people to not violate implicit societal expectations.
• Second-order punishment (anger at those who are accepting of transgressor) incentivize a community to respond to, and develop a unified response to, dyadic disagreements.

Takeaways

• In Propriety Frames, we saw how the brain retains social information.
• Limbic machinery, such as the Care module, generate normative intuitions.
• Generators and frames interact to facilitate both top-down and bottom-up learning.
• A subset of social violations are tagged as morally intolerable.
• Several emotion devices work to consolidate normatives within a group.

Related Work

• Graham, Haidt, Nosek (2009). Liberals and conservatives rely on different sets of moral foundations.
• Cushman, Young, Greene (2010). Our multi-system moral psychology: Towards a consensus view
• Tetlock et al (2000). The psychology of the unthinkable: taboo trade-offs, forbidden base rates, and heretical counterfactuals.

# [Sequence] Psychology Musings

This sequence is simply a bucket that collects some of my older, “pre-theoretic” views on psychology. YMMV. 🙂

Random musings

Books I enjoyed

# [Sequence] Cognitive Architecture

Main Sequence

Version Iterations

• Version 0.1: The mind as a collection of functional modules.
• Version 1.0: Integrating anatomy and genetics.
• Version 1.1: Integrating ecology, morality, frame memory, and social systems.
• Version 1.2: Integrating theory of mind, social reasoning and culture.
• Version 1.3: Integrating memory hierarchy, motivation, frame memory.
• Version 1.4: Integrating working memory, attention, consciousness.
• Version 1.5: Integrating cognitive control, affective neuroscience.

# Singular Value Decomposition

Part Of: Algebra sequence
Followup To: Eigenvalues and Eigenvectors
Content Summary: 1300 words, 13 min read.

Limitations of Eigendecomposition

Last time, we learned how to locate eigenvalues and eigenvectors of a given matrix. Diagonalization is the process where a square matrix can be decomposed into two factors: a matrix $Q$ (with eigenvectors along the columns) and a matrix $\Lambda$ (with eigenvalues along the diagonal).

$A = Q \Lambda Q^T$

But as we saw with the spectral theorem, eigendecomposition only works well against square, symmetric matrices. If a matrix isn’t symmetric, it is easy to run into complex eigenvalues. And if a matrix isn’t square, you are out of luck entirely!

Can we generalize eigendecomposition to apply to a wider family of matrices? Can we diagonalize matrices of any size, even if they don’t have “nice” properties?

Yes. Self-transposition is the key insight of our “eigendecomposition 2.0”. We define the self-transpositions of $A$ as $AA^{T}$ and $A^{T}A$.

Suppose $A \in \mathbb{R}^{m x n}$. Then $AA^T \in \mathbb{R}^{mxm}$ and $A^TA \in \mathbb{R}^{nxn}$. So these matrices are square. But they are also symmetric!

To illustrate, consider the following.

$A = \begin{bmatrix} 4 & 4 \\ -3 & 3 \\ \end{bmatrix}$

Since A is not symmetric, we have no guarantee that its eigenvalues are real. Indeed, its eigenvalues turn out to be complex:

$\det(A - \lambda I) = \begin{bmatrix} 4 - \lambda & 4 \\ -3 & 3 - \lambda \\ \end{bmatrix} = 0$

$(12 - 7 \lambda + \lambda^2) + 12 = 0 \Rightarrow \lambda^2 -7 \lambda + 24 = 0$

$\lambda = \frac{7 \pm \sqrt{(-7)^2 - 4*1*24}}{2*1} = \frac{7}{2} \pm \frac{\sqrt{47}i}{2}$

Eigendecomposition on $A$ sucks. Are the self-transposed matrices any better?

$A^TA = \begin{bmatrix} 4 & -3 \\ 4 & 3 \\ \end{bmatrix} \begin{bmatrix} 4 & 4 \\ -3 & 3 \\ \end{bmatrix} = \begin{bmatrix} 25 & 7 \\ 7 & 25 \\ \end{bmatrix}$

$AA^T = \begin{bmatrix} 4 & 4 \\ -3 & 3 \\ \end{bmatrix} \begin{bmatrix} 4 & -3 \\ 4 & 3 \\ \end{bmatrix} = \begin{bmatrix} 32 & 0 \\ 0 & 18 \\ \end{bmatrix}$

These matrices are symmetric! Thus, they are better candidates for eigendecomposition.

Towards Singular Value Decomposition

Singular Value Decomposition (SVD) is based on the principle that all matrices are eigendecomposable after self-transposition. It is essentially a bug fix:

An important way to picture SVD is with the idea of orthogonal bases. It is relatively easy to find any number of orthogonal bases for a given rowspace. Call the matrix of orthogonal vectors $V$.

We desire to find an orthogonal basis $V$ such that $AV$ produces an orthogonal basis in the column space. Orthogonal bases are not particularly hard to find. But most orthogonal bases, once projected to column space, will lose their orthogonality! We desire that particular orthogonal basis $U$ such that $V$ is also orthogonal.

We won’t require vectors in $U$ to be the same size as those in $V$. Instead, we will normalize $V$; its basis vectors will be orthonormal (orthogonal and normal). Then the length of each vector in $U$ will diverge by a scaling factor.

As we will soon see, these scaling factors are not eigenvalues. Instead, we will use sigmas instead of lambdas.

• Scaling factors $\sigma$ , analogous to eigenvalues $\lambda$.
• Diagonal matrix $\Sigma$ , analogous to diagonal matrix $\Lambda$

Our full picture then, looks like this:

Let us now translate this image into matrix language.

$A \begin{bmatrix} \vdots & \vdots & \vdots & \vdots \\ v_1 & v_2 & \dots & v_n \\ \vdots & \vdots & \vdots & \vdots \\ \end{bmatrix} = \begin{bmatrix} \vdots & \vdots & \vdots & \vdots \\ \sigma_1u_1 & \sigma_2u_2 & \dots & \sigma_nu_n \\ \vdots & \vdots & \vdots & \vdots \\ \end{bmatrix}$

But we can easily factorize the right-hand side:

$\begin{bmatrix} \vdots & \vdots & \vdots & \vdots \\ \sigma_1u_1 & \sigma_2u_2 & \dots & \sigma_nu_n \\ \vdots & \vdots & \vdots & \vdots \\ \end{bmatrix} = \begin{bmatrix} \vdots & \vdots & \vdots & \vdots \\ u_1 & u_2 & \dots & u_n \\ \vdots & \vdots & \vdots & \vdots \\ \end{bmatrix} * \begin{bmatrix} \sigma_1 & 0 & \dots & 0 \\ 0 & \sigma_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \sigma_n \\ \end{bmatrix}$

So we have that:

$AV = U \Sigma$

Since both $V$ and $U$ are orthogonal, inversion of either is equivalent to transposition:

$A = U \Sigma V^{-1} = U \Sigma V^T$

This strongly resembles our diagonalization equation $A = Q \Lambda Q^T$. SVD distinguishes itself by considering two orthogonal eigenmatrices $U$ and $V$, not just one ($Q$).

Recalling that $A = U \Sigma V^T$ ,

$A^TA = (V \Sigma^T U^T) (U \Sigma V^T)$

But now the innermost term cancels. Since $\Sigma$ is a square diagonal matrix, its self-transposition is simply equal to $\Sigma^{2}$. So,

$A^TA = V \Sigma^2 V^T$

Since $A^{T}A$ is a square, symmetric matrix, our diagonalization theorem applies!

$A^TA = V \Sigma ^2 V^T = Q \Lambda Q^T$

To find $U$, a similar trick works:

$AA^T = (U \Sigma V^T)(V \Sigma^T U^T) = U \Sigma^2 U^T = Q \Lambda Q^T$

The relationships between SVD and eigendecomposition are as follows:

• $V$ is the eigenvectors of $A^TA$
• $U$ is the eigenvectors of $AA^T$
• $\Sigma$ is the square root of the eigenvalues matrix $\Lambda$

If any eigenvalue is negative, the corresponding sigma factor would be complex. But $A^TA$ and $AA^T$ are positive-semidefinite, which guarantees non-negative eigenvalues. This assures us that $\Sigma$ contains only real values.

In contrast to eigendecomposition, every matrix has an SVD decomposition. Geometrically, $V$ and $U$ act as rotational transformations, and Sigma acts as a scaling transformation. In other words, every linear transformation comprises a rotation, then scaling, then another rotation.

A Worked Example

Let’s revisit $A$. Recall that:

$A = \begin{bmatrix} 4 & 4 \\ -3 & 3 \\ \end{bmatrix}, A^TA = \begin{bmatrix} 25 & 7 \\ 7 & 25 \\ \end{bmatrix}, AA^T = \begin{bmatrix} 32 & 0 \\ 0 & 18 \\ \end{bmatrix}$

Eigendecomposition against $A$ is unpleasant because $A$ is not symmetric. But $A^{T}A$ is guaranteed to be positive semi-definite; that is, to have non-negative eigenvalues. Let’s see this in action.

$det(A^TA - \lambda I) = \begin{bmatrix} 25 - \lambda & 7 \\ 7 & 25 - \lambda \\ \end{bmatrix} = 0$

$\lambda^2 - 50 \lambda + 576 = 0 \Rightarrow (\lambda_1 - 32)(\lambda_2 - 18) = 0$

$\lambda_1 = 32, \lambda_2 = 18$

$trace(A^TA) = 50 = \sum{\lambda_i}$

$det(A^TA) = 625-49 = 576 = 18 * 32 = \prod{\lambda_i}$

These are positive, real eigenvalues. Perfect! Let’s now derive the corresponding (normalized) eigenvectors.

$A - 32I = \begin{bmatrix} -7 & 7 \\ 7 & -7 \\ \end{bmatrix} = 0, A - 18I = \begin{bmatrix} 7 & 7 \\ 7 & 7 \\ \end{bmatrix} = 0$

$rref(A - 32I) = \begin{bmatrix} -1 & 1 \\ 0 & 0 \\ \end{bmatrix} = 0, rref(A-18I) = \begin{bmatrix} 1 & 1 \\ 0 & 0 \\ \end{bmatrix} = 0$

$v_1 = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \\ \end{bmatrix}, v_2 = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} \\ \end{bmatrix}$

SVD intends to decompose $A$ into $U \Sigma V^{T}$. The above findings give us two of these ingredients.

$V^{T} = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \end{bmatrix}$

$\Sigma =\begin{bmatrix} \sqrt{32} & 0 \\ 0 & \sqrt{18} \\ \end{bmatrix}$

What’s missing? $U$! To find it, we perform eigendecomposition on $AA^{T}$. This is an especially easy task, because $AA^{T}$ is already a diagonal matrix.

$AA^T = \begin{bmatrix} 32 & 0 \\ 0 & 18 \\ \end{bmatrix}$

$U = \begin{bmatrix} u_1 & u_2 \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}$

We have arrived at our first Singular Value Decomposition.

$A = U \Sigma V^T = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} \sqrt{32} & 0 \\ 0 & \sqrt{18} \\ \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \\ \end{bmatrix}$

Okay, so let’s check our work. 😛

$A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \left( \begin{bmatrix} \sqrt{32} & 0 \\ 0 & \sqrt{18} \\ \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \\ \end{bmatrix} \right) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} 4 & 4 \\ 3 & 3 \\ \end{bmatrix} = \begin{bmatrix} 4 & 4 \\ 3 & 3 \\ \end{bmatrix}$

These matrices are a viable factorization: multiplication successfully recovers $A$.

Takeaways

• Eigendecomposition only works for a subclass of matrices; SVD decomposes all matrices.
• SVD relies on self-transposition to convert any arbitrary matrix into one that works well against eigendecomposition (guarantees square $m = n$ and symmetric $A = A^{T}$).
• Another way to interpret SVD is by taking a special kind of orthogonal basis that, once passed through the linear transformation, preserves its orthogonality.
• Every matrix $A = U \Sigma V^{T}$. That is, every linear transformation can be conceived as rotation + scaling + rotation.

Until next time.

# Eigenvalues and Eigenvectors

Part Of: Algebra sequence
Followup To: An Introduction to Linear Algebra
Next Up: Singular Value Decomposition
Content Summary: 1300 words, 13 min read

Geometries of Eigenvectors

Matrices are functions that act on vectors, by mapping from row-vectors to column-vectors.  Consider two examples:

1. Reflection matrices, which reflect vectors across some basis.
2. Rotation matrices, which rotate vectors clockwise by $\theta$ degrees.

The set of eigenvectors of a matrix $A$ is a special set of input vectors for which the matrix behaves as a scaling transformation. In other words, we desire the set of vectors $\vec{x}$ whose output vectors $A\vec{x}$ differ by a scaling factor.

Eigenvectors have a straightforward geometric interpretation:

1. Reflection eigenvectors are orthogonal or parallel to the reflecting surface. In the left image above, that is the top two pairs of vectors.
2. Rotation eigenvectors do not exist (more formally, cannot be visualized in $\mathbb{R}^2$).

Algebra of Eigenvectors

We can express our “parallel output” property as:

$A\vec{x} = \lambda \vec{x}$

Thus $\vec{x}$ and $A\vec{x}$ point in the same direction, but differ by scaling factor $\lambda$.

Scaling factor $\lambda$ is the eigenvalue. There can be many $\left( x, \lambda \right)$ pairs that satisfy the above equality.

For an $\mathbb{R}^{n x n}$ matrix, there are $n$ eigenvalues. These eigenvalues can be difficult to find. However, two facts aid our search:

• The sum of eigenvalues equals the trace (sum of values along the diagonal).
• The product of eigenvalues equals the determinant.

To solve, subtract $\lambda \vec{x}$ from both sides:

$A\vec{x} = \lambda \vec{x}$

$(A - \lambda I)\vec{x} = 0$

We would like to identify n unique eigenvectors. But if the new matrix $(A - \lambda I)$ has an empty nullspace, it will contain zero eigenvectors. So we desire this new matrix to be singular.

How to accomplish this?  By finding eigenvalues that satisfy the characteristic equation $\det(A - \lambda I) = 0$. Matrices are singular iff their determinants equal zero.

Let’s work through an example! What is the eigendecompositon for matrix $A$:

$A = \begin{bmatrix} 3 & 1 \\ 1 & 3 \\ \end{bmatrix}$

We need to find eigenvalues that solve the characteristic equation.

$\det(A - \lambda I) = \begin{vmatrix} 3-\lambda & 1 \\ 1 & 3-\lambda \\ \end{vmatrix} = 0$

$(3 - \lambda)^2 - 1^2 = \lambda_2 -6\lambda + 8 = (\lambda-2)(\lambda-4) = 0$

$\lambda_1 = 2, \lambda_2 = 4$

Are these eigenvalues correct? Let’s check our work:

$trace(A) = 6 = \sum{\lambda_i}$

$det(A) = 8 = \prod{\lambda_i}$

How to find our eigenvectors? By solving the nullspace given each eigenvalue.

For $\lambda_1=2$ :

$A - 2I = \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix} \Rightarrow rref(A - 2I) = \begin{bmatrix} 1 & 1 \\ 0 & 0 \\ \end{bmatrix}$

$(\lambda_1, \vec{x}_1) = (2, \begin{bmatrix} 1 \\ -1 \\ \end{bmatrix})$

For $\lambda_2=4$ :

$A - 4I = \begin{bmatrix} -1 & 1 \\ 1 & -1 \\ \end{bmatrix} \Rightarrow rref(A - 4I) = \begin{bmatrix} -1 & 1 \\ 0 & 0 \\ \end{bmatrix}$

$(\lambda_2, \vec{x}_2) = (4, \begin{bmatrix} 1 \\ 1 \\ \end{bmatrix})$

Desirable Matrix Properties

The above example was fairly straightforward. But eigendecomposition can “go awry”, as we shall see. Consider a rotation matrix, which in two dimensions has the following form:

$R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \\ \end{bmatrix}$

What are the eigenvalues for rotation $\theta = 90^{\circ}$ ?

$R = \begin{bmatrix} 0 & -1 \\ 1 & 0 \\ \end{bmatrix}$

$\det(R - \lambda I) = \begin{bmatrix} - \lambda & -1 \\ 1 & - \lambda \\ \end{bmatrix} = 0$

$(- \lambda)^2 - 1^2 \Rightarrow \lambda^2 = 1$

$\lambda_1 = i, \lambda_2 = -i$

We can check our work:

$trace(R) = 0 = \sum{\lambda_i}$

$det(R) = 1 = \prod{\lambda_i}$

We saw earlier that rotation matrices have no geometric interpretation. Here, we have algebraically shown that its eigenvalues are complex.

$A = \left[ \begin{smallmatrix} 3 & 1 \\ 1 & 3 \\ \end{smallmatrix} \right]$ has real eigenvalues, but $R = \left[ \begin{smallmatrix} 0 & -1 \\ 1 & 0 \\ \end{smallmatrix} \right]$ has less-desirable complex eigenvalues.

We can generalize the distinction between $A$ and $R$ as follows:

Spectral Theorem. Any matrix that is symmetric (A = AT) is guaranteed to have real, nonnegative eigenvalues. The corresponding n eigenvectors are guaranteed to be orthogonal.

In other words, eigendecomposition works best against symmetric matrices.

Diagonalization

Let us place each eigenvector in the column of a matrix $S$. What happens when you multiply the original matrix $A$ by this new matrix? Since $S$ contains eigenvectors, multiplication by $A$ reduces to multiplication by the associated eigenvalues:

$AS = \begin{bmatrix} \vdots & \vdots & \vdots & \vdots \\ \lambda_1x_1 & \lambda_2x_2 & \dots & \lambda_nx_n \\ \vdots & \vdots & \vdots & \vdots \\ \end{bmatrix}$

We see the product contains a mixture of eigenvalues and eigenvectors. We can separate these by “pulling out” the eigenvalues into a diagonal matrix. Call this matrix $\Lambda$ (“capital lambda”).

$AS = \begin{bmatrix} \vdots & \vdots & \vdots & \vdots \\ x_1 & x_2 & \dots & x_n \\ \vdots & \vdots & \vdots & \vdots \\ \end{bmatrix} * \begin{bmatrix} \lambda_1 & 0 & \dots & 0 \\ 0 & \lambda_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \lambda_n \\ \end{bmatrix} = S \Lambda$

Most matrices have the property that its eigenvectors are linearly independent. For such matrices, $S$ is invertible. Given this fact, we can solve for $A$:

$\Lambda = S^{-1} A S$

$A = S \Lambda S^{-1}$

Matrices that can be factorized in this way are said to be diagonalizable. We can see that both elimination and eigendecomposition are performing the same type of work: factorizing matrices into their component parts.

If $A$ is symmetric, then we know $\Lambda$ is real, and its eigenvectors in $S$ are orthogonal. Let us rename $S$ to be $Q$, to reflect this additional property. But orthogonal matrices have the property that transposition equats inversion: $Q^T = Q^{-1}$. Thus, if $A$ is symmetric, we can simplify the diagonalization formula to:

$A = Q \Lambda Q^{-1} = Q \Lambda Q^T$

Asymptoptic Interpretations

This diagonalization approach illustrates an important use case of eigenvectors: power matrices. What happens when $A$ is applied arbitrarily many times? What does the output look like in the limit?

We can use the diagonalization equation to represent $A^k$:

$A^k = \prod_{i=1}^{k} (Q^{-1} \Lambda Q) = (Q^{-1} \Lambda Q)(Q^{-1} A Q)\dots(Q^{-1} \Lambda Q)$

We can simplify by canceling the inner terms $QQ^{-1}$:

$A^k = Q^{-1} \Lambda^k Q$

This equation tells us that the eigenvectors is invariant to how many times $A$ is applied. In contrast, eigenvalue matrix $\Lambda$ has important implications for ongoing processes:

• If each eigenvalue has magnitude less than one, the output will trend towards zero.
• If each eigenvalue has magnitude greater than one, the output will trend to infinity.

Fibonacci Eigenvalues

The powers interpretation of eigenvalues sheds light on the behavior of all linear processes. This includes number sequences such as the Fibonacci numbers, where each number is the sum of the previous two numbers.

Recall the Fibonacci numbers are $0,1,1,2,3,5,8,13,...$ What is $F_{100}$ ?

Eigenvalues can answer this question. We must first express the Fibonacci generator as a linear equation:

$F(k+2) = 1F(k+1) + 1F(k)$

In order to translate this into a meaningful matrix, we must add a “redundant” equation

$F(k+1) = 1F(k+1) + 0F(k)$

With these equations, we can create a 2×2 Fibonacci matrix $F$.

$F = \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ \end{bmatrix}$

This matrix uniquely generates Fibonacci numbers.

$u_1 = Fu_0 = \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ \end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ \end{bmatrix}$

$u_4 = F^4u_0 = \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ \end{bmatrix}^4 \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix} = \begin{bmatrix} 5 \\ 3 \\ \end{bmatrix}$

To discover the rate at Fibonacci numbers grow, we decompose $F$ into its eigenvalues:

$\det(F - \lambda I) = \begin{vmatrix} 1 - \lambda & 1 \\ 1 & 1- \lambda \\ \end{vmatrix} = 0$

$\lambda^2 - 2\lambda - 1 = 0$

$\lambda_1 = \frac{1 + \sqrt{5}}{2}, \lambda_2 = \frac{1 - \sqrt{5}}{2}$

$\lambda_1 = 1.61803, \lambda_2 = -0.61803$

$trace(F) = 1 = \sum{\lambda_i}$

$det(F) = -1 = \prod{\lambda_i}$

We can go on to discover eigenvectors $x_1$ and $x_2$. We can then express the Fibonnaci matrix $F$ as

$F = \lambda_1x_1 + \lambda_2 x_2$

$F^k = \lambda_1^k x_1 + \lambda_2^k x_2$

As k goes to infinity, the second term goes to zero. Thus, the ratio is dominated by the larger eigenvalue, 1.61803.

Mathematicians in the audience will recognize this number as the golden ratio.

We have long known that the ratio of successive Fibonnaci numbers converges to 1.61803. Eigenvalues provide a mechanism to derive this value analytically.

Until next time.


text:
# Linear Approximations with a hemispherical dome

1. Dec 11, 2011

### forestmine

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

Use linear approximations to estimate the amount of paint needed to apply a
coat of pain .05cm thick to a hemispherical dome with diameter 50m.

2. Relevant equations

V=2/3*$\pi$*r$^{3}$

3. The attempt at a solution

I'm really not at all sure how to go about this. The only thing I somewhat understand is that it's necessary to subtract the difference in volume to obtain just the volume of the shell itself.

If someone could walk me through this one, that'd be great.

Thanks!

Last edited: Dec 11, 2011


text:
Comment on On Discriminative vs. Generative Classifiers: A Comparison of Logistic Regression and Naive Bayes"
Jinghao Xue and Mike Titterington
Neural Processing Letters 2007.

## Abstract

Comparison of generative and discriminative classifiers is an ever-lasting topic. Based on their theoretical and empirical comparisons between the na\"{i}ve Bayes classifier and linear logistic regression, Ref.~\citeauthor{Ng:2001} claimed that there existed two distinct regimes of performance between the generative and discriminative classifiers with regard to the training-set size. However, our empirical and simulation studies, as presented in this paper, suggest that it is not so reliable to claim such an existence of the two distinct regimes. In addition, for real world datasets, so far there is no theoretically correct, general criterion for choosing between the discriminative and the generative approaches to classification of an observation $\mathbf{x}$ into a class $y$; the choice depends on the relative confidence we have in the correctness of the specification of either $p(y|\mathbf{x})$ or $p(\mathbf{x}, y)$. This can be to some extent a demonstration of why Ref.~\citeauthor{Efron:1975} and~\citeauthor{ONeill:1980} prefer LDA but other empirical studies may prefer linear logistic regression instead. Furthermore, we suggest that pairing of either LDA assuming a common diagonal covariance matrix (LDA-$\Lambda$) or the na\"{i}ve Bayes classifier and linear logistic regression may not be perfect, and hence it may not be reliable for any claim that was derived from the comparison between LDA-$\Lambda$ or the na\"{i}ve Bayes classifier and linear logistic regression to be generalised to all the generative and discriminative classifiers.


text:
# Compoundoku wants BIG

Rules of Compoundoku:

• Solve both left and right Sudokus.
• In addition, the board below them is the Compound Board of both Sudokus.
• Each number on the Compound Board should tell either: (1) the number on the left Sudoku, or (2) the sum of both numbers on the left and right Sudokus; in the respective position.

The original and easier version can be found here.
(Adding this as a note to avoid the duplicate warning when posting.)

I believe this is the solution :)

Some information on how I solved it:

The bottom two 10s and 11s are key. Obviously these can't be the left sodoku values since the largest value is 6. So they must be a sum. The top 10 also can't the sum of two 5s otherwise the 11 left of it will have two 5's in the same row. Putting that all together gives you twelve starting values. Then the 9 must be the sum of 6 and 3, left and right respectively, otherwise it will break the rules of the local sodokus. I think I made an early guess that of the three 2s, the middle one was an actual 2 which turned out correctly. I think the next biggest clue was the 5s. The rest started to fall into place from there fairly quickly. Great puzzle by the way.

• Can you explain why? For those who don't get it? – Mukyuu Jan 17 at 4:50
• Yes, just added some explanation. Thought I should get my solution in before I get sniped :) – Earlien Jan 17 at 4:52
• Yup, this is correct! Well done! – athin Jan 17 at 10:35

There is no need to guess, also proves that solution is unique. L stands for left sudoku, R for the right one. (X,Y) means the X is in L, Y in R, the rest should be self-evident. Formatting of sudokus is crappy but I believe it should be clear enough.

First solve 10s and 11s. Note that 11 is 6+5, this prevents 10s in same rows (= all of them) from being 5+5. Additionally, focus on 10 in the second bottom row - that one has to be 4 in L because 5 and 6 are taken by 11s. This quickly gives solution for all 10 and 11 for both sudokus.

Then

Fill in 6 in the R, using ordinary sudoku rules. This is exactly on the position of left column top 2 of the combined sudoku, this also gives 1,1 for the other 2s. At this point we have the following image:

L/R, horizontal lines are not drawn.

x5x|6xx . . . x6|x4|xx
xxx|xxx . . . xx|xx|xx
x1x|xxx . . . x1|xx|xx
2xx|xxx . . . 6x|xx|xx
xxx|4x6 . . . xx|x6|x5
1x6|x5x . . . 1x|4x|6x

Then

Top left 5 can't be 1,2 or 5 because of L, it cannot be 1 in R, so it only has (3,2) option left. This enables filling in 5 in bottom left rectangle of the L, using ordinary sudoku rules, followed by 4 in same rectangle, 6 in top-left one, then 6 in mid left, 3 in mid left, 2 and 3 in bottom left, 4 in top left.

Solution of L/R at this point is

45x|6xx . . . x6|x4|xx
36x|xxx . . . xx|xx|xx
61x|xxx . . . x1|xx|xx
23x|xxx . . . 6x|xx|xx
523|4x6 . . . xx|x6|x5
146|x5x . . . 1x|4x|6x

Now notice that

2 is in position of 5 of combined sudoku, giving 3 for the R. The other 5 in the 2nd row cannot be 3,4 in L and 2,4 in R, so it has to be (2,3). Fill in the remaining 1 in top-left of L. The 2nd row 4 has to be 4 of the L, because 2 and 3 are blocked in both sudokus, preventing 1+3 or 2+2 combination. Then 9 has to be (6,3) because all other numbers from the L are taken.

At this point we have

451|6xx . . . x6|x4|xx
362|x4x . . . 2x|3x|xx
61x|xxx . . . x1|xx|xx
23x|x6x . . . 6x|xx|3x
523|4x6 . . . x3|x6|x5
146|x5x . . . 1x|4x|6x

Note

Top right 4 of combined sudoku has to be composite, and cannot be (1,3) because of L. This means 3 in top right of R has only one spot left. Fill in the other 3s. Then finish left column of R, then 4, bottom right 1 and 2, 2nd left column and remaining numbers.

This gives

451|6xx . . . 36|54|21
362|x4x . . . 24|31|56
61x|xxx . . . 51|62|43
23x|x6x . . . 62|15|34
523|4x6 . . . 43|26|15
146|x5x . . . 15|43|62

Only one part left now:

Top right 4 is composite and now we know it is 3 in L. This gives all other 3, and 2 in the top row with 1 in the bottom right part, then 2 in the bottom right part, 2 in mid right part. Now we know that the last number 5 of combined sudoku has to refer to the left one, because the remaining option (2,3) is not possible. This now solves the remainder of the sudoku (start with 5, then 4, then 1).

Solution is, as Earlien already found:

451|623 . . . 36|54|21
362|541 . . . 24|31|56
614|235 . . . 51|62|43
235|164 . . . 62|15|34
523|416 . . . 43|26|15
146|352 . . . 15|43|62

• This is an impressive work! :o Well done btw! – athin Jan 17 at 15:28


text:
# Is there any way to check whether a group is residually solvable?

For a given group presentation of a group(finitely presented), I want to check whether it is residually solvable or not. Is there any good way to do it?

Actually, I'm curious whether the finitely presented group $G = < x_i, 1\leq i \leq m | w_i x_i w_i^{-1}x_{i+1}, 1\leq i \leq n-1 >$, where $w_i=x_j^{\pm 1}$ for some $j$.

-
@HJ: There are misprints in the definition of $G$. Could you please correct them? Is it supposed to be an LOG presentation? –  Mark Sapir Dec 8 '10 at 21:36
HJ - as stated, this is just a free group. (Killing a conjugate of $x_i$ is the same as killing $x_i$.) I presume you're aware of Magnus's Theorem that free groups are residually nilpotent (in particular, residually solvable). –  HJRW Dec 8 '10 at 21:42
@Henry, it is a clear misprint. He probably meant an LOG presentation. –  Mark Sapir Dec 8 '10 at 21:53
@Mark, Yes, I corrected the presentation. There were missing $x_[i+1}$. –  hjjang Dec 9 '10 at 20:29

This problem is undecidable. If it were not, you could use it to construct an algorithm for testing if a given f.p. group is trivial or not, which is well known to be undecidable:

 Input: f.p. group G

 Test if G is residually solvable. If it is not, output "non-trivial". If it is, find the abelianization of G. If the abelianization is trivial, output "trivial".
• Otherwise output "non-trivial".
-
Let me add that decidability of this problem is still open for one-relator groups: maths.tcd.ie/pub/ims/bull65/S6501.pdf. –  Jon Bannon Dec 8 '10 at 13:14

It is not clear whether the set of finite residually solvable group presentations is even recursively enumerable. Unlike for the word or triviality problem where there exists an algorithm which says "yes" iff the answer is "yes", I do not think there exists such an algorithm in this case. But I do not think anybody proved that the algorithm does not exist. Same for residually finite groups.

-


text:
# Round Me, Help Me

Given an input n, your program or function must output the smallest positive integer k such that n rounded to the nearest multiple of k is greater than n.

### Example.

Given an input 20, the output value should be 3:

• The nearest multiple of 1 is 20, which is not greater than 20.

• The nearest multiple of 2 is 20, which is not greater than 20.

• The nearest multiple of 3 is 21, which is greater than 20, so it is output.

### Test Cases

#Input  #Output
2       3
4       5
6       4
8       3
10      4
12      7
14      3
16      6
18      4
20      3
22      4
24      5
26      3
28      5
30      4
32      3
34      4
36      8
38      3
40      6
42      4
44      3
46      4
48      5
50      3
52      6
54      4
56      3
58      4
60      7
62      3
64      5
66      4
68      3
70      4
72      11
74      3
76      6
78      4
80      3
82      4
84      5
86      3
88      5
90      4
92      3
94      4
96      7
98      3
1000    6

The output given any odd input should be 2.

### Rules

• n is a positive integer less than 2^32
• rounding is performed such that if two multiples of k are equally distant from n, the greater one is chosen ("round halves up"). In this way, every odd n yields an output of 2.
• This is , so shortest code in each language wins.
• I've edited the format of your test cases to make it easier to read, and more concise. Let me know if you have any problems with this, or if any of the new examples are off. :) – James Aug 16 '17 at 17:58
• @Shaggy Done! I removed 500 odds and 450 evens from the list. – fireflame241 Aug 16 '17 at 19:08
• Is there an oeis link for this sequence? – James K Aug 17 '17 at 22:45
• @JamesK I did not find one when I searched earlier. Possibly someone with an OEIS account could make one? – fireflame241 Aug 17 '17 at 22:49

# Python 3, 48 38 bytes

Edit: -10 bytes by using recursion

l=lambda x,y=2:y*(x%y>=y/2)or l(x,y+1)

Try it online!

# Japt, 6 bytes

@<rX}a

Try it online!

## Explanation:

@    <r X}a
XYZ{U<UrX}a
X              // X = 0; Increments when the condition in between {...} fails
{     }a    // Return the first integer X where:
U          //   The input
<U        //   is less than the input
rX      //     rounded to the nearest multiple of X

• r is a builtin? o_o – Erik the Outgolfer Aug 16 '17 at 18:46
• @EriktheOutgolfer: Japt also has built-ins for rounding up or down :) – Shaggy Aug 16 '17 at 18:55
• I knew this feature would come in handy someday :D – ETHproductions Aug 16 '17 at 19:59
• @Shaggy that's nuts! o_o_o – Erik the Outgolfer Aug 17 '17 at 8:52
• @Oliver: This has me more convinced to get to grips with function methods, now - my own version of this was 7 bytes: o æ@<rX – Shaggy Aug 17 '17 at 10:59

# MATL, 13 bytes

tQ:yy/Yo*<fX<

### Explanation

Consider input 6.

t      % Implicit input. Duplicate
% STACK: 6, 6
% STACK: 6, [1 2 3 4 5 6 7]
yy     % Duplicate top two elements
% STACK: 6, [1 2 3 4 5 6 7], 6, [1 2 3 4 5 6 7]
/      % Divide, element-wise
% STACK: 6, [1 2 3 4 5 6 7], [6 3 2 1.5 1.2 1 0.8571]
Yo     % Round to closest integer. Halves are rounded up
% STACK: 6, [1 2 3 4 5 6 7], [6 3 2 2 1 1 1]
*      % Multiply, element-wise
% STACK: 6, [6 6 6 8 5 6 7]
<      % Less than, element-wise
% STACK: [0 0 0 1 0 0 1]
f      % Find: indices of nonzeros (1-based)
% STACK: [4 7]
X<     % Minimum of vector. Implicit display
% STACK: 4

# Python 2, 35 bytes

f=lambda n,i=1:n%i*2/i or-~f(n,i+1)

Try it online!

# JavaScript (ES6), 28 25 bytes

n=>g=x=>n%x>=x/2?x:g(-~x)

• 3 bytes saved thanks to Arnauld.

## Test it

o.innerText=(f=

n=>g=x=>n%x>=x/2?x:g(-~x)

)(i.value=64)();oninput=_=>o.innerText=f(+i.value)()
<input id=i type=number>

Or test all numbers from 1-1000 (Give it a minute to run):

f=n=>g=x=>n%x>=x/2?x:g(-~x);[2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,8,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,7,2,3,2,5,2,4,2,3,2,4,2,11,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,11,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,8,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,7,2,3,2,5,2,4,2,3,2,4,2,13,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,9,2,3,2,5,2,4,2,3,2,4,2,8,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,8,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,7,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,10,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,11,2,3,2,5,2,4,2,3,2,4,2,8,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,8,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,9,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,7,2,3,2,5,2,4,2,3,2,4,2,8,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,8,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,10,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,7,2,3,2,5,2,4,2,3,2,4,2,8,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,8,2,3,2,5,2,4,2,3,2,4,2,9,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,10,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,7,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,8,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,8,2,3,2,5,2,4,2,3,2,4,2,13,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,13,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,8,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,7,2,3,2,5,2,4,2,3,2,4,2,11,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,6,2,4,2,3,2,4,2,9,2,3,2,5,2,4,2,3,2,4,2,7,2,3,2,6,2,4,2,3,2,4,2,5,2,3,2,5,2,4,2,3,2,4,2,8,2,3,2,6].forEach((x,y)=>o.innerText+=\n-----------------------------\n${(++y+"").padStart(7)} |${(y=f(y)()+"").padStart(8)}  |  ${x==y})
 Input | Output | Pass
 # Proton, 33 bytes n=>[x for x:2..n+2if n%x>=x/2][0]  Try it online! • I don't know anything about Proton, but It seems you can save 3 bytes: Try it online! – jferard Aug 17 '17 at 10:25 • Maybe a coincidence, but this is exactly the same as totallyhuman's solution...:p – Erik the Outgolfer Aug 18 '17 at 16:29 • @EriktheOutgolfer We posted it at the same time (in fact I ninja'd him by a few seconds) with 37-byter, because Hyper borked the operators, and when he fixed them we both updated. – Mr. Xcoder Aug 18 '17 at 16:30 • Uhh, I ninja'd you IIRC. :P – totallyhuman Aug 18 '17 at 16:38 • @totallyhuman You ninja'd me with a 41-byter. I posted the 37-byter first and ninja'd you with that one by a few seconds. – Mr. Xcoder Aug 18 '17 at 16:41 # Pyth, 12 11 bytes hfgy%QTTm+2  Try it here. # Proton, 33 bytes x=>[y for y:2..x+2if x%y>=y/2][0]  Try it online! • FWIW, why did you remove ? – Mr. Xcoder Aug 18 '17 at 16:43 # Jelly, 11 bytes ÷R%1<.¬;1TṂ  A monadic link taking and returning positive integers. Try it online! or see a test suite. ### How? ÷R%1<.¬;1TṂ - Link: number, n e.g. 10 R - range(n) [ 1,2,3 ,4 ,5,6 ,7 ,8 ,9 ,10] ÷ - n divided by [10,5,3.33..,2.5,2,1.66..,1.42..,1.25,1.11..,1 ] %1 - modulo by 1 [ 0,0,0.33..,0.5,0,0.66..,0.42..,0.25,0.11..,0 ] <. - less than 0.5? [ 1,1,1 ,0 ,1,0 ,1 ,1 ,1 ,1 ] ¬ - not [ 0,0,0 ,1 ,0,1 ,0 ,0 ,0 ,0 ] ;1 - concatenate a 1 [ 0,0,0 ,1 ,0,1 ,0 ,0 ,0 ,0 , 1] T - truthy indices [ 4 ,6 ,11] Ṃ - minimum 4  Note: The concatenation of 1 is just to handle the cases where n is one of 1 ,2, or 4 when the result needs to be n+1 (‘R÷@%1<.¬TṂ would also work). # Haskell, 33 32 bytes f n=[i|i<-[1..],2*mod n i>=i]!!0  Try it online! Saved one byte thanks to w0lf • You can use !!0 instead of head – Cristian Lupascu Aug 17 '17 at 8:28 # Dyalog APL, 23 22 bytes {⊃x/⍨⍵<x×⌊.5+⍵÷x←⍳1+⍵}  Try it online! # Pyth, 5 bytes fgy%Q  Test suite No rounding builtins, just checking for the first positive integer T, where double the input mod T is greater than or equal to T. Explanation: fgy%Q fgy%QTT Implicit variable introduction. f Find the first positive integer T such that the following is truthy: %QT Input % T y Doubled g T Is greater than or equal to T  # x86 Machine Code, 17 bytes This code implements a basic, iterative solution in the form of a reusable function: 31 F6 xor esi, esi 46 inc esi ; set ESI (our temp register) to 1 Loop: 89 C8 mov eax, ecx ; copy 'n' to EAX for division 46 inc esi ; eagerly increment temp 99 cdq ; extend EAX into EDX:EAX F7 F6 div esi ; divide EDX:EAX by ESI 01 D2 add edx, edx ; multiply remainder by 2 39 F2 cmp edx, esi ; compare remainder*2 to temp 7C F4 jb Loop ; keep looping if remainder*2 < temp 96 xchg eax, esi ; put result into EAX (1 byte shorter than MOV) C3 ret  The function follows the fastcall calling convention, so that the single parameter (n) is passed in the ECX register. The return value (k) is, per usual, returned in the EAX register. Try it online! # Java 8, 42 bytes Lambda from Integer to Integer. n->{for(int f=1;;)if(n%++f*2>=f)return f;}  Try It Online ## Acknowledgments • -1 byte thanks to Kevin Cruijssen • You can save a byte by starting f=1 and using ++f on the first f, like this: n->{for(int f=1;;)if(n%++f*2>=f)return f;} – Kevin Cruijssen Aug 17 '17 at 7:54 # Perl 5, 24 + 1 (-p) = 25 bytes 1while$_%++$k<$k/2;$_=$k

Try it online!

Tries each integer $k starting at 1 until it finds a remainder that is at least half of $k.

# Forth (gforth), 45 bytes

: f 1 begin 1+ 2dup mod over 1+ 2/ >= until ;

Try it online!

### Code Explanation

: f             \ start a new word definition
1             \ start a counter at 1
begin         \ start an indefinite loop
1+          \ add 1 to counter
2dup mod    \ duplicate input value and counter, get remainder of input/counter
over 1+ 2/  \ get counter/2 (add 1 to force rounding up)
>=          \ check if remainder is greater than counter/2
until         \ end loop if true, otherwise go back to beginning
;               \ end word definition

# 05AB1E, 9 bytes

∞.ΔIs/Dò‹

Try it online!

## Explanation

∞.ΔIs/Dò‹ Full code
∞.Δ       Returns the first number for which the following code returns true
-> stack is [n]
Is     Push the input and swap the stack -> stack is [input, n]
/    Divide both of them -> stack is [input/n]
Dò  Duplicate and round the second -> stack is [input/n, rounded(input/n)]
‹ Check if input/n got larger by rounding -> stack is [bool]
-> if bool is true, abort and return the current number

# Rockstar, 681 bytes

Thought takes Patience and Control
While Patience is as high as Control
Let Patience be without Control

Give back Patience

Rock takes Art
Love is neverending
Sex is bottomless
Give back Art
Else
Limits are inspiration
Give back the rubbish

Listen to Chance
Questions are unstoppable
Until Questions is Chance
Build Questions up
Put Thought taking Chance, Questions into your mind
Put Questions over Answers into the world
Put Rock taking the world into the world
If your mind is as big as the world
Say Questions
Break it down

You can try rockstar online, but you'll need to copy and paste the code across. It will prompt you for an input number.

I didn't go for lowest byte count , because Rockstar is obviously not made for golfing, so instead I tried to go for Rock 'n' Roll lyrics.

## Explanation:

This is based on the same solution as others (python, java):

Iterate up from 2:
if n % iterator >= ceil(n/2)
return iterator

First I need to define modulus and ceiling functions though, which for poetry's sake are called Thought and Rock.

The below is a less poetic version with different variable names, and explanations where the syntax is unclear. Parentheses denote comments.

Modulus takes Number and Divisor
While Number is as high as Divisor
Put Number minus Divisor into Number
(blank line ending While block)
Give back Number (return Number)
(blank line ending function declaration)
Ceil takes Decimal
Put Modulus taking Decimal, 1 into Remainder
If Remainder is 0
Give back Decimal (return Decimal)
Else
Put Decimal with 1 minus Remainder into Result
Give back Result (return Result)
(blank line ending if block)
(blank line ending function declaration)
Listen to Input (Read from STDIN to Input)
Index is 1
Until Index is Input
Build Index up (Increment by 1)
Put Modulus taking Input, Index into LHS
Put Index over 2 into RHS
Put Ceil taking RHS into RHS
If LHS is as big as RHS
Say Index
Break it down (Break from loop)

ɓ÷Ḟ,¥Ċ$×ạÐṂ⁸Ṁ>⁸µ1#  Try it online! Full program. • Full program. When is it not? – totallyhuman Aug 16 '17 at 18:57 • @totallyhuman Sometimes it also works as a function. – Erik the Outgolfer Aug 17 '17 at 8:53 # Swift 3, 51 bytes {n in(2..<n+2).filter{Float(n%$0)>=Float($0)/2}[0]}  For some extremely bizarre reasons, [0] doesn't work online. Here is the online compiler-compatible version (that uses .first! instead): {n in(2..<n+2).filter{Float(n%$0)>=Float(\$0)/2}.first!}

Test Suite (online-compatible).

# C# (Mono), 39 bytes

n=>{int i=1;for(;n%++i*2<i;);return i;}

Try it online!

# Stax, 7 bytes

ç&╬▼⌐╔ñ

Run and debug it