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IMR Press / JIN / Volume 21 / Issue 5 / DOI: 10.31083/j.jin2105129
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Open Access Original Research
Gait Disorders and Magnetic Resonance Imaging Characteristics in Older Adults with Cerebral Small Vessel Disease
Ping Zhao1,†Yumeng Gu1,†Xin Li1,*
Show Less
1 Department of Neurology, Second Hospital of Tianjin Medical University, 300211 Tianjin, China
2 Department of Rehabilitation, Second Hospital of Tianjin Medical University, 300211 Tianjin, China
*Correspondence: lixinlx9935@126.com (Xin Li)
These authors contributed equally.
J. Integr. Neurosci. 2022, 21(5), 129; https://doi.org/10.31083/j.jin2105129
Submitted: 8 February 2022 | Revised: 24 March 2022 | Accepted: 31 March 2022 | Published: 21 July 2022
This is an open access article under the CC BY 4.0 license.
Abstract

Background: This study aims to explore the features of gait disorders with cerebral small vessel disease (CSVD), and results from magnetic resonance imaging (MRI) with diffusion tensor imaging (DTI) were analyzed. Methods: The 139 patients with CSVD were divided into two groups by the Tinetti scale scores: the gait disorder (GD) group with a score $<$24 (63 patients) and the normal gait (GN) group with a score $\geq$24 (76 patients). A series of scales and 3.0T MRI with DTI were used to analyze the correlation between the abnormal gaits and imaging findings. Results: The differences in the Barthel Index, Unified Parkinson’s Disease Rating Scale part III, and Montreal Cognitive Assessment (MoCA) scores between the two groups were significant (p $<$ 0.05), and there were significant correlations between MoCA and total gait scores (r = 0.201, p = 0.002). The GD group had a more degraded gait score, widened gait base, and degraded gait length than the GN group (p $<$ 0.05). There were significant differences between the two groups (p $<$ 0.05) in white matter (WM) hyperintensities (WMH) of the Fazekas scale grade 2–3 and lacunes. The GD group had a greater total MRI burden than the GN group (p $<$ 0.05). In DTI parameters, the GD group had lower fractional anisotropy (FA) and higher mean diffusion (MD) values in WM tracts in many areas around the ventricles (family-wise error corrected, p $<$ 0.05). Significant correlations were observed between FA and the total gait score (r = 0.467, p $<$ 0.01), and also between MD and total gait score (r  = –0.422, p $<$ 0.01). Conclusions: Patients with CSVD with gait disorders had more WMH of Fazekas scale grade 2–3, lacunes, and total MRI burden than the GN patients, and those with gait disorders may suffer from demyelination of nerve fibers and damage to the fibers’ microstructures.

Keywords
cerebral small vessel disease
gait disorder
WMH
lacunes
CMB
EPVS
total MRI burden
DTI
Figures
Fig. 1.
Share


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# GLPK/Using the GLPK callable library

This page describes issues related to the use of GLPK as a callable library. As such this page should be primarily of interest to developers writing application programs in C or C++ and making use of the GLPK application programming interface (API).

## Official documentation

The official GLPK documentation file doc/glpk.pdf contains a full reference for the GLPK API. That material is therefore not repeated here. See obtaining GLPK.

## Short example

The following API example implements the linear constrained optimization model:

 maximize $0.6 x_1 + 0.5 x_2$ subject to $x_1 + 2 x_2 \le 1$ $3 x_1 + x_2 \le 2$
```/* short.c */

#include <stdio.h>            /* C input/output                       */
#include <stdlib.h>           /* C standard library                   */
#include <glpk.h>             /* GNU GLPK linear/mixed integer solver */

int main(void)
{
/* declare variables */
glp_prob *lp;
int ia[1+1000], ja[1+1000];
double ar[1+1000], z, x1, x2;
/* create problem */
lp = glp_create_prob();
glp_set_prob_name(lp, "short");
glp_set_obj_dir(lp, GLP_MAX);
/* fill problem */
glp_set_row_name(lp, 1, "p");
glp_set_row_bnds(lp, 1, GLP_UP, 0.0, 1.0);
glp_set_row_name(lp, 2, "q");
glp_set_row_bnds(lp, 2, GLP_UP, 0.0, 2.0);
glp_set_col_name(lp, 1, "x1");
glp_set_col_bnds(lp, 1, GLP_LO, 0.0, 0.0);
glp_set_obj_coef(lp, 1, 0.6);
glp_set_col_name(lp, 2, "x2");
glp_set_col_bnds(lp, 2, GLP_LO, 0.0, 0.0);
glp_set_obj_coef(lp, 2, 0.5);
ia[1] = 1, ja[1] = 1, ar[1] = 1.0; /* a[1,1] = 1 */
ia[2] = 1, ja[2] = 2, ar[2] = 2.0; /* a[1,2] = 2 */
ia[3] = 2, ja[3] = 1, ar[3] = 3.0; /* a[2,1] = 3 */
ia[4] = 2, ja[4] = 2, ar[4] = 1.0; /* a[2,2] = 1 */
glp_load_matrix(lp, 4, ia, ja, ar);
/* solve problem */
glp_simplex(lp, NULL);
/* recover and display results */
z = glp_get_obj_val(lp);
x1 = glp_get_col_prim(lp, 1);
x2 = glp_get_col_prim(lp, 2);
printf("z = %g; x1 = %g; x2 = %g\n", z, x1, x2);
/* housekeeping */
glp_delete_prob(lp);
glp_free_env();
return 0;
}
```

Production code would, of course, check the return value from function glp_simplex and react accordingly.

### Linux

These instructions relate to Linux (the details for other operating systems should be similar).

Build short using your system compiler, in this case GNU GCC:

```\$ gcc -Wall short.c -lglpk -o short
```

Run the resulting binary:

```\$ ./short
*     0: obj =   0.000000000e+00  infeas =  0.000e+00 (0)
*     2: obj =   4.600000000e-01  infeas =  0.000e+00 (0)
OPTIMAL SOLUTION FOUND
z = 0.46; x1 = 0.6; x2 = 0.2
```

You should obtain $x_1 = 0.6$ and $x_2 = 0.2$ as part of the terminal output from your program.

The GLPK runtime reporting format (the lines starting *) are covered in the callable library part of the official GLPK documentation and also here.

The same problem is coded in MathProg.

## Submitting patches

GLPK patches are best submitted by posting them to the most appropriate of the two GLPK mailing lists. Note that the GLPK project does not maintain a web-based source code repository.

If creating source file patches, please refrain from using tabs and restrict line lengths to 72 chars.

GLPK is not inherently thread safe. Please consult the GLPK newsgroup archives for more information about the use of GLPK APIs under concurrent execution. In particular, see this discussion and, more latterly, this discussion.

If your problem involves multiple LPs (or can be structured thus), then consider running each LP as a separate processes. This makes sense only for multi-core hardware.

## Reenterability

Andrew said that reenterability is a property of the code that allows running several instances of the same program in parallel using the only copy of that program in the memory. This property requires the program not to alter its own code and statically allocated data. See also Wikipedia definition of reentrancy.

The glpk code is reenterable except two internal routines tls_set_ptr and tls_get_ptr (see src/glpenv02.c). In a reenterable version of glpk these two routines should be replaced with platform-specific ones.

The current version of glpenv02.c is not reentrant:

```static void *tls = NULL;

void tls_set_ptr(void *ptr)
{     tls = ptr;
return;
}

void *tls_get_ptr(void)
{     void *ptr;
ptr = tls;
return ptr;
}
```

Andrew suggested reentrant version:

```#include <pthread.h>
/* must be initialized in the main program */

void tls_set_ptr(void *ptr)
return;
}

void *tls_get_ptr(void)
{     void *ptr;
return ptr;
}
```

Latest GCC compilers have storage class keyword __thread :

```static __thread void *tls = NULL;

void tls_set_ptr(void *ptr)
{     tls = ptr;
}

void *tls_get_ptr(void)
{     return tls;
}
```

Visual Studio 2005 compiler has similar storage-class modifier __declspec( thread ):

```static __declspec( thread ) void *tls = NULL;

void tls_set_ptr(void *ptr)
{     tls = ptr;
}

void *tls_get_ptr(void)
{     return tls;
}
```

## Deprecated GLPK APIs

A number of GLPK APIs have been recently deprecated and replaced. Unless otherwise stated, the lpx_ prefix has been replaced with a glp_ prefix. In all cases, developers should double check with the official documentation.

As of version 4.49, developers must migrate to these new APIs. The old API routines, whose names begin with lpx_, have now been removed from the GLPK API and are no longer available.

GLPK Deprecated Comment
lpx_create_index
lpx_create_prob
lpx_del_cols
lpx_del_rows
lpx_delete_index
lpx_delete_prob
lpx_find_col
lpx_find_row
lpx_get_col_dual
lpx_get_col_kind
lpx_get_col_lb
lpx_get_col_name
lpx_get_col_prim
lpx_get_col_stat
lpx_get_col_type
lpx_get_col_ub
lpx_get_mat_col
lpx_get_mat_row
lpx_get_num_bin
lpx_get_num_cols
lpx_get_num_int
lpx_get_num_nz
lpx_get_num_rows
lpx_get_obj_coef
lpx_get_obj_dir
lpx_get_obj_name
lpx_get_obj_val
lpx_get_prob_name
lpx_get_row_dual
lpx_get_row_lb
lpx_get_row_name
lpx_get_row_prim
lpx_get_row_stat
lpx_get_row_type
lpx_get_row_ub
lpx_ipt_col_dual
lpx_ipt_col_prim
lpx_ipt_obj_val
lpx_ipt_row_dual
lpx_ipt_row_prim
lpx_mip_col_val
lpx_mip_obj_val
lpx_mip_row_val
lpx_set_col_bnds
lpx_set_col_kind
lpx_set_col_name
lpx_set_col_stat
lpx_set_mat_col
lpx_set_mat_row
lpx_set_obj_coef
lpx_set_obj_dir
lpx_set_obj_name
lpx_set_prob_name
lpx_set_row_bnds
lpx_set_row_name
lpx_set_row_stat
4.18 lpx_simplex
4.20 lpx_integer see: glp_iotopt
lpx_write_mps
lpx_write_freemps
lpx_write_cpxlp
4.31 lpx_scale_prob
lpx_std_basis
lpx_cpx_basis
4.32 lpx_integer
lpx_intopt
4.33 lpx_exact
lpx_get_ray_info see: glp_unbnd_ray
lpx_interior
4.37 lpx_print_sol
lpx_print_ips see: lpx_print_ipt
lpx_print_mip
lpx_print_prob see: glp_print_lp
4.41 lpx_transform_row
lpx_transform_col
lpx_prim_ratio_test
lpx_dual_ratio_test
4.42 lpx_print_sens_bnds see: glp_print_ranges
4.49 lpx_check_kkt see: glp_check_kkt

## Recently added GLPK APIs

New APIs get added to GLPK from time to time. These are listed below. If appropriate, a new API may also be exposed by an associated GLPSOL command-line option in the same or a subsequent release.

Note: this information only dates back as far as release 4.31 in February 2007.

4.49 glp_check_kkt replaces lpx_check_kkt
glp_mincost_relax4 solve minimum cost flow problem using RELAX-IV code
4.47 glp_intfeas1 solve CNF-SAT problem instance with GLPK code (tentative)
4.46 glp_read_cnfsat read CNF-SAT problem data in DIMACS format
glp_write_cnfsat write CNF-SAT problem data in DIMACS format
glp_check_cnfsat check CNF-SAT problem instance
glp_minisat1 solve CNF-SAT problem instance with MiniSat
4.44 glp_cpp solve critical path problems
4.42 glp_check_dup check for duplicate elements in sparse matrix
glp_sort_matrix sort elements of the constraint matrix
glp_read_prob read problem data in GLPK format
glp_write_prob write problem data in GLPK format
glp_analyze_bound analyze active bound of non-basic variable
glp_analyze_coef analyze objective coefficient at basic variable
glp_print_ranges print sensitivity analysis report
4.41 glp_transform_row transform explicitly specified row
glp_transform_col transform explicitly specified column
glp_prim_rtest perform primal ratio test
glp_dual_rtest perform dual ratio test
4.40 glp_del_vertices remove vertices from graph
glp_del_arc remove arc from graph
glp_wclique_exact find maximum weight clique with the exact algorithm developed by Prof P Ostergard
glp_read_ccdata read graph in DIMACS clique/coloring format
glp_write_ccdata write graph in DIMACS clique/coloring format
4.39 glp_warm_up "warm up" LP basis
glp_set_vertex_name assign (change) vertex name
glp_create_v_index create vertex name index
glp_find_vertex find vertex by its name
glp_delete_v_index delete vertex name index
glp_read_asnprob read assignment problem data in DIMACS format
glp_write_asnprob write assignment problem data in DIMACS format
glp_check_asnprob check correctness of assignment problem data
glp_asnprob_lp convert assignment problem to LP
glp_asnprob_okalg solve assignment problem with the out-of-kilter algorithm
glp_asnprob_hall find bipartite matching of maximum cardinality with Hall's algorithm
4.37 glp_print_sol write basic solution in printable format
glp_print_ipt write interior-point solution in printable format
glp_print_mip write MIP solution in printable format
glp_read_graph read (di)graph from plain text file
glp_write_graph write (di)graph to plain text file
glp_weak_comp find all weakly connected components
glp_strong_comp find all strongly connected components
4.36 glp_mincost_okalg find minimum-cost flow with out-of-kilter algorithm
glp_maxflow_ffalg find maximal flow with Ford-Fulkerson algorithm
4.35 glp_create_graph create graph
glp_set_graph_name assign (change) graph name
glp_erase_graph erase graph content
glp_delete_graph delete graph
glp_read_mincost read minimum cost flow problem data in DIMACS format
glp_write_mincost write minimum cost flow problem data in DIMACS format
glp_mincost_lp convert minimum cost flow problem to LP
glp_netgen Klingman's network problem generator
glp_gridgen grid-like network problem generator
glp_read_maxflow read maximum flow problem data in DIMACS format
glp_write_maxflow write maximum flow problem data in DIMACS format
glp_maxflow_lp convert maximum flow problem to LP
glp_rmfgen Goldfarb's maximum flow problem generator
4.33 glp_copy_prob copy problem object content
glp_exact solve LP in exact arithmetic (makes lpx_exact deprecated)
glp_get_unbnd_ray determine variable causing unboundedness (makes lpx_get_ray_info deprecated)
glp_interior solve LP with interior-point method (makes lpx_interior deprecated)
4.32 glp_ios_row_attr determine additional row attributes
glp_ios_pool_size determine current size of the cut pool
glp_ios_add_row add constraint to the cut pool
glp_ios_del_row delete constraint from the cut pool
glp_ios_clear_pool delete all constraints from the cut pool
4.31 glp_scale_prob automatic scaling of problem data
glp_std_basis construct standard initial LP basis

format: plain


text:
# How to add special character in LaTeX?

In my current work I need to use arrow symbols, representing injection, surjection and bijection. First two can be given by commands \rightarrowtail and \twoheadrightarrow, but the bijection arrow - Unicode 2916 character - is not present in the list of LaTeX arrow characters. How can I add the mentioned symbol in my text ? Maybe some additional packages should be used ?

I'm new to LaTeX, so any help will be appreciated. Thanks in advance.

-
Have a look at “How to look up a math symbol?” on tex.stackexchange.com for ideas how you can easily find a particular LaTeX symbol. –  Martin Scharrer Jul 8 '11 at 19:56

I can emulate it like this :

\documentclass{minimal}

\usepackage{amssymb}

\begin{document}

$\rightarrowtail\!\!\!\!\!\rightarrow$

\end{document}

I would also advise looking at the Comprehensive LaTeX symbol List in general.

-
+1 for symbols-a4.pdf –  Svante Jul 8 '11 at 13:58

Using the great Detexify2 - LaTeX symbol classifier tool I found \twoheadrightarrow and \rightarrowtail which both combined build the wanted arrow. You can use TeX macro \kern to adjust the kerning between the symbols. However, I found that it doesn't produce the correct alignment in all size (e.g. \tiny uses different versions of the symbols), so be careful. The combination should be placed inside \mathrel{..} to get the correct surrounding spacing to the symbols before and after.

\documentclass{article}

\usepackage{amssymb}

%\newcommand*{\twoheadrightarrowtail}{\mathrel{\rlap{$\rightarrowtail$}\twoheadrightarrow}}
$A \twoheadrightarrowtail B$


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# Compound Inequalities

## Key Questions

• The same way you would solve normal inequalities, only there are more sides to the inequality now. remember: what you do to one side you do it ALL sides

1.-3≤2x-1<5

2.-3+1≤2x-1+1<5+1

3.-2/2≤(2x)/2<6/2

4.-1≤x<3


text:
# zbMATH — the first resource for mathematics

## Netuka, Ivan

Compute Distance To:
 Author ID: netuka.ivan Published as: Netuka, Ivan; Netuka, I. Homepage: https://www2.karlin.mff.cuni.cz/~netuka/indexan.html External Links: MGP · ORCID · Wikidata · ResearchGate · dblp · GND · IdRef
 Documents Indexed: 110 Publications since 1971, including 2 Books 6 Contributions as Editor Reviewing Activity: 14 Reviews Biographic References: 1 Publication
all top 5

#### Co-Authors

 37 single-authored 25 Hansen, Wolfhard 24 Veselý, Jiří 15 Lukes, Jaroslav 3 Bečvářová, Martina 3 Gardiner, Stephen J. 2 Anděl, Jiří 2 Kurzweil, Jaroslav 2 Malý, Jan 2 Schwabik, Štefan 2 Spurný, Jiří 1 Aleksandrov, Pavel Sergeevich 1 Armitage, David H. 1 Bauer, Heinz 1 Berg, Christian 1 Bliedtner, Jürgen 1 Chatterji, Srishti Dhar 1 Feyel, Denis 1 Frolík, Zdeněk 1 Fuchs, Eduard 1 Ghergu, Marius 1 Goldstein, Myron 1 GowriSankaran, Kohur N. 1 Hayman, Walter Kurt 1 Heyer, Herbert 1 Jacob, Niels 1 Klima, Vlastimil 1 Kučera, Petr 1 Lions, Pierre-Louis 1 Manolaki, Myrto 1 Mocek, Tomáš 1 Munkholm, Hans Jørgen 1 Nečas, Jindřich 1 O’Farrell, Anthony G. 1 Ranocha, Pavel 1 Render, Hermann 1 Salinger, David L. 1 Sanabria-García, M. A. 1 Smrčka, Michael 1 Soucek, Vladimir 1 Zajíček, Luděk 1 Zvára, Karel
all top 5

#### Serials

 18 Pokroky Matematiky, Fyziky & Astronomie 8 Časopis Pro Pěstování Matematiky 8 Czechoslovak Mathematical Journal 6 Commentationes Mathematicae Universitatis Carolinae 6 Mathematische Annalen 6 Expositiones Mathematicae 5 Potential Analysis 4 Mathematica Bohemica 3 Advances in Mathematics 3 Journal of Approximation Theory 2 American Mathematical Monthly 2 Journal d’Analyse Mathématique 2 Journal of Mathematical Analysis and Applications 2 Proceedings of the American Mathematical Society 1 Archive for History of Exact Sciences 1 Israel Journal of Mathematics 1 Mathematical Proceedings of the Cambridge Philosophical Society 1 Revue Roumaine de Mathématiques Pures et Appliquées 1 Bulletin des Sciences Mathématiques. Deuxième Série 1 Kybernetika 1 Mathematische Zeitschrift 1 Osaka Journal of Mathematics 1 Proceedings of the London Mathematical Society. Third Series 1 Studii și Cercetări Matematice 1 Statistics 1 Journal of Theoretical Probability 1 Acta Universitatis Matthiae Belii. Series Mathematics 1 Journal of Convex Analysis 1 Bulletin des Sciences Mathématiques 1 Annales Academiae Scientiarum Fennicae. Mathematica 1 Positivity 1 De Gruyter Studies in Mathematics 1 Lecture Notes in Mathematics 1 NATO ASI Series. Series C. Mathematical and Physical Sciences 1 Publications in Sciences. University of Joensuu 1 Supplemento ai Rendiconti del Circolo Matemàtico di Palermo. Serie II 1 Dějiny Matematiky / History of Mathematics 1 Revista Brasileira de História da Matemática 1 Analysis and Mathematical Physics 1 Heritage of European Mathematics
all top 5

#### Fields

 75 Potential theory (31-XX) 27 History and biography (01-XX) 19 Partial differential equations (35-XX) 15 Functional analysis (46-XX) 12 Probability theory and stochastic processes (60-XX) 8 Functions of a complex variable (30-XX) 6 General and overarching topics; collections (00-XX) 5 Real functions (26-XX) 5 Measure and integration (28-XX) 4 Operator theory (47-XX) 4 General topology (54-XX) 3 Approximations and expansions (41-XX) 3 Convex and discrete geometry (52-XX) 2 Statistics (62-XX) 1 Mathematical logic and foundations (03-XX) 1 Number theory (11-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Integral equations (45-XX) 1 Global analysis, analysis on manifolds (58-XX)

#### Citations contained in zbMATH Open

53 Publications have been cited 240 times in 164 Documents Cited by Year
Mean value property and harmonic functions. Zbl 0863.31012
Netuka, Ivan; Veselý, Jiří
1994
Integral representation theory. Applications to convexity, Banach spaces and potential theory. Zbl 1216.46003
Lukeš, Jaroslav; Malý, Jan; Netuka, Ivan; Spurný, Ji{ří}
2010
On threshold autoregressive processes. Zbl 0547.62058
Anděl, Jiří; Netuka, Ivan; Zvára, Karel
1984
The third boundary value problem in potential theory. Zbl 0242.31007
Netuka, Ivan
1972
Generalized Robin problem in potential theory. Zbl 0241.31008
Netuka, Ivan
1972
The Dirichlet problem for harmonic functions. Zbl 0454.31002
Netuka, Ivan
1980
An operator connected with the third boundary value problem in potential theory. Zbl 0241.31009
Netuka, Ivan
1972
On approximation of affine Baire-one functions. Zbl 1031.35011
Lukeš, J.; Malý, J.; Netuka, I.; Smrčka, M.; Spurný, J.
2003
Unavoidable sets and harmonic measures living on small sets. Zbl 1307.31012
Hansen, Wolfhard; Netuka, Ivan
2014
Volume densities with the mean value property for harmonic functions. Zbl 0818.31003
Hansen, W.; Netuka, I.
1995
On the Picard principle for $$\Delta + \mu$$. Zbl 1248.31004
Hansen, Wolfhard; Netuka, Ivan
2012
Fredholm radius of a potential theoretic operator for convex sets. Zbl 0314.31006
Netuka, Ivan
1975
Convexity properties of harmonic measures. Zbl 1146.31006
Hansen, Wolfhard; Netuka, Ivan
2008
Jensen measures in potential theory. Zbl 1250.31004
Hansen, Wolfhard; Netuka, Ivan
2012
Inverse mean value property of harmonic functions. Zbl 0794.31001
Hansen, Wolfhard; Netuka, I.
1993
The Robin problem in potential theory. (Preliminary communication). Zbl 0215.42602
Netuka, I.
1971
Potential theory of the farthest point distance function. Zbl 1134.31001
Gardiner, Stephen J.; Netuka, Ivan
2007
Representation of potentials. Zbl 1389.31018
Hansen, Wolfhard; Netuka, Ivan
2014
Continuity and maximum principle for potentials of signed measures. Zbl 0309.31019
Netuka, Ivan
1975
Smooth surfaces with infinite cyclic variation. Zbl 0204.08002
Netuka, I.
1971
Elliptic points in one-dimensional harmonic spaces. Zbl 0222.31018
Král, Josef; Lukeš, Jaroslav; Netuka, Ivan
1971
Champagne subdomains with unavoidable bubbles. Zbl 1288.31001
Hansen, Wolfhard; Netuka, Ivan
2013
Thinness and the heat equation. Zbl 0289.35037
Netuka, Ivan
1974
Contractivity of C. Neumann’s operator in potential theory. Zbl 0372.31004
Kral, Josef; Netuka, Ivan
1977
Hunt’s hypothesis (H) and triangle property of the Green function. Zbl 1338.31018
Hansen, Wolfhard; Netuka, Ivan
2016
Scaling invariant Harnack inequalities in a general setting. Zbl 1381.60109
Hansen, Wolfhard; Netuka, Ivan
2016
Cluster sets of harmonic measures and the Dirichlet problem in potential theory. Zbl 0759.31004
Netuka, Ivan
1992
Potential theory, surveys and problems. Proceedings of a conference held in Prague, July 19-24, 1987. Zbl 0642.00008
1988
Choquet’s theory and the Dirichlet problem. Zbl 1046.31001
Lukeš, Jaroslav; Netuka, Ivan; Veselý, Jiří
2002
Double layer potential representation of the solution of the Dirichlet problem (Preliminary communication). Zbl 0255.31009
Netuka, Ivan
1973
Double layer potentials and the Dirichlet problem. Zbl 0308.31008
Netuka, Ivan
1974
Fine behaviour of solutions of the Dirichlet problem near an irregular point. Zbl 0699.31015
Netuka, Ivan
1990
A Liouville theorem for nonlinear elliptic systems with isotropic nonlinearities. Zbl 0517.35034
Lions, Pierre-Louis; Nečas, Jindřich; Netuka, Ivan
1982
Density of extremal measures in parabolic potential theory. Zbl 1191.31003
Hansen, Wolfhard; Netuka, Ivan
2009
The classical Dirichlet problem and its generalizations. Zbl 0432.31013
Netuka, Ivan
1980
Harmonic measures for a point may form a square. Zbl 1204.30016
Hansen, Wolfhard; Netuka, Ivan
2010
Approximation by harmonic functions and the Dirichlet problem. Zbl 0759.41033
Netuka, I.
1992
Locally uniform approximation by solutions of the classical Dirichlet problem. Zbl 0773.31010
Hansen, W.; Netuka, I.
1993
Measure and topology: Mařík spaces. Zbl 0879.28029
Netuka, I.
1996
In memoriam Professor Jan Mařík (1920-1994). Zbl 0801.01014
Král, Josef; Kurzweil, Jaroslav; Netuka, Ivan; Schwabik, Štefan
1994
Reduced functions and Jensen measures. Zbl 1380.31002
Hansen, Wolfhard; Netuka, Ivan
2018
Methods for calculating stationary distribution in linear models of time series. Zbl 1120.62070
Anděl, J.; Netuka, I.; Ranocha, P.
2007
Nearly hyperharmonic functions and Jensen measures. Zbl 1421.31009
Hansen, Wolfhard; Netuka, Ivan
2019
Pervasive function spaces and the best harmonic approximation. Zbl 0641.41025
Netuka, Ivan
1987
Fuchs, Eduard; Netuka, Ivan
1988
Smoothness of a typical convex function. Zbl 0499.26004
Klima, Vlastimil; Netuka, Ivan
1981
Some properties of potentials of signed measures. Zbl 0289.31009
Netuka, Ivan
1974
The Wiener type solution of the Dirichlet problem in potential theory. Zbl 0319.31009
Lukes, Jaroslav; Netuka, Ivan
1976
Harmonic continuation and removable singularities in the axiomatic potential theory. Zbl 0358.31007
Netuka, Ivan; Vesely, Jiri
1978
The third boundary value problem in potential theory. Zbl 0383.31002
Netuka, Ivan
1975
Successive averages and harmonic functions. Zbl 0879.31002
Hansen, W.; Netuka, Ivan
1997
The farthest point distance function. Zbl 1147.30300
Gardiner, Stephen J.; Netuka, Ivan
2007
Karl Löwner and his student Lipman Bers. Pre-war Prague mathematicians. Zbl 1320.01001
Bečvářová, Martina; Netuka, Ivan
2015
Nearly hyperharmonic functions and Jensen measures. Zbl 1421.31009
Hansen, Wolfhard; Netuka, Ivan
2019
Reduced functions and Jensen measures. Zbl 1380.31002
Hansen, Wolfhard; Netuka, Ivan
2018
Hunt’s hypothesis (H) and triangle property of the Green function. Zbl 1338.31018
Hansen, Wolfhard; Netuka, Ivan
2016
Scaling invariant Harnack inequalities in a general setting. Zbl 1381.60109
Hansen, Wolfhard; Netuka, Ivan
2016
Karl Löwner and his student Lipman Bers. Pre-war Prague mathematicians. Zbl 1320.01001
Bečvářová, Martina; Netuka, Ivan
2015
Unavoidable sets and harmonic measures living on small sets. Zbl 1307.31012
Hansen, Wolfhard; Netuka, Ivan
2014
Representation of potentials. Zbl 1389.31018
Hansen, Wolfhard; Netuka, Ivan
2014
Champagne subdomains with unavoidable bubbles. Zbl 1288.31001
Hansen, Wolfhard; Netuka, Ivan
2013
On the Picard principle for $$\Delta + \mu$$. Zbl 1248.31004
Hansen, Wolfhard; Netuka, Ivan
2012
Jensen measures in potential theory. Zbl 1250.31004
Hansen, Wolfhard; Netuka, Ivan
2012
Integral representation theory. Applications to convexity, Banach spaces and potential theory. Zbl 1216.46003
Lukeš, Jaroslav; Malý, Jan; Netuka, Ivan; Spurný, Ji{ří}
2010
Harmonic measures for a point may form a square. Zbl 1204.30016
Hansen, Wolfhard; Netuka, Ivan
2010
Density of extremal measures in parabolic potential theory. Zbl 1191.31003
Hansen, Wolfhard; Netuka, Ivan
2009
Convexity properties of harmonic measures. Zbl 1146.31006
Hansen, Wolfhard; Netuka, Ivan
2008
Potential theory of the farthest point distance function. Zbl 1134.31001
Gardiner, Stephen J.; Netuka, Ivan
2007
Methods for calculating stationary distribution in linear models of time series. Zbl 1120.62070
Anděl, J.; Netuka, I.; Ranocha, P.
2007
The farthest point distance function. Zbl 1147.30300
Gardiner, Stephen J.; Netuka, Ivan
2007
On approximation of affine Baire-one functions. Zbl 1031.35011
Lukeš, J.; Malý, J.; Netuka, I.; Smrčka, M.; Spurný, J.
2003
Choquet’s theory and the Dirichlet problem. Zbl 1046.31001
Lukeš, Jaroslav; Netuka, Ivan; Veselý, Jiří
2002
Successive averages and harmonic functions. Zbl 0879.31002
Hansen, W.; Netuka, Ivan
1997
Measure and topology: Mařík spaces. Zbl 0879.28029
Netuka, I.
1996
Volume densities with the mean value property for harmonic functions. Zbl 0818.31003
Hansen, W.; Netuka, I.
1995
Mean value property and harmonic functions. Zbl 0863.31012
Netuka, Ivan; Veselý, Jiří
1994
In memoriam Professor Jan Mařík (1920-1994). Zbl 0801.01014
Král, Josef; Kurzweil, Jaroslav; Netuka, Ivan; Schwabik, Štefan
1994
Inverse mean value property of harmonic functions. Zbl 0794.31001
Hansen, Wolfhard; Netuka, I.
1993
Locally uniform approximation by solutions of the classical Dirichlet problem. Zbl 0773.31010
Hansen, W.; Netuka, I.
1993
Cluster sets of harmonic measures and the Dirichlet problem in potential theory. Zbl 0759.31004
Netuka, Ivan
1992
Approximation by harmonic functions and the Dirichlet problem. Zbl 0759.41033
Netuka, I.
1992
Fine behaviour of solutions of the Dirichlet problem near an irregular point. Zbl 0699.31015
Netuka, Ivan
1990
Potential theory, surveys and problems. Proceedings of a conference held in Prague, July 19-24, 1987. Zbl 0642.00008
1988
Fuchs, Eduard; Netuka, Ivan
1988
Pervasive function spaces and the best harmonic approximation. Zbl 0641.41025
Netuka, Ivan
1987
On threshold autoregressive processes. Zbl 0547.62058
Anděl, Jiří; Netuka, Ivan; Zvára, Karel
1984
A Liouville theorem for nonlinear elliptic systems with isotropic nonlinearities. Zbl 0517.35034
Lions, Pierre-Louis; Nečas, Jindřich; Netuka, Ivan
1982
Smoothness of a typical convex function. Zbl 0499.26004
Klima, Vlastimil; Netuka, Ivan
1981
The Dirichlet problem for harmonic functions. Zbl 0454.31002
Netuka, Ivan
1980
The classical Dirichlet problem and its generalizations. Zbl 0432.31013
Netuka, Ivan
1980
Harmonic continuation and removable singularities in the axiomatic potential theory. Zbl 0358.31007
Netuka, Ivan; Vesely, Jiri
1978
Contractivity of C. Neumann’s operator in potential theory. Zbl 0372.31004
Kral, Josef; Netuka, Ivan
1977
The Wiener type solution of the Dirichlet problem in potential theory. Zbl 0319.31009
Lukes, Jaroslav; Netuka, Ivan
1976
Fredholm radius of a potential theoretic operator for convex sets. Zbl 0314.31006
Netuka, Ivan
1975
Continuity and maximum principle for potentials of signed measures. Zbl 0309.31019
Netuka, Ivan
1975
The third boundary value problem in potential theory. Zbl 0383.31002
Netuka, Ivan
1975
Thinness and the heat equation. Zbl 0289.35037
Netuka, Ivan
1974
Double layer potentials and the Dirichlet problem. Zbl 0308.31008
Netuka, Ivan
1974
Some properties of potentials of signed measures. Zbl 0289.31009
Netuka, Ivan
1974
Double layer potential representation of the solution of the Dirichlet problem (Preliminary communication). Zbl 0255.31009
Netuka, Ivan
1973
The third boundary value problem in potential theory. Zbl 0242.31007
Netuka, Ivan
1972
Generalized Robin problem in potential theory. Zbl 0241.31008
Netuka, Ivan
1972
An operator connected with the third boundary value problem in potential theory. Zbl 0241.31009
Netuka, Ivan
1972
The Robin problem in potential theory. (Preliminary communication). Zbl 0215.42602
Netuka, I.
1971
Smooth surfaces with infinite cyclic variation. Zbl 0204.08002
Netuka, I.
1971
Elliptic points in one-dimensional harmonic spaces. Zbl 0222.31018
Král, Josef; Lukeš, Jaroslav; Netuka, Ivan
1971
all top 5

#### Cited by 158 Authors

 26 Netuka, Ivan 24 Hansen, Wolfhard 15 Spurný, Jiří 12 Medková, Dagmar 7 Verbitsky, Igor E. 6 Rondoš, Jakub 5 Volchkov, Vitaliĭ Vladimirovich 4 Hamhalter, Jan 4 Lanconelli, Ermanno 4 Lukes, Jaroslav 4 Turilova, Ekaterina A. 3 Cupini, Giovanni 3 Frazier, Michael W. 3 Hu, Ze-Chun 3 Loperfido, Nicola M. R. 3 Sun, Wei 3 Volchkov, Valeriĭ Vladimirovich 3 Vondraček, Zoran 2 Adamowicz, Tomasz 2 Alakhrass, Mohammad 2 Anděl, Jiří 2 Brzezina, Miroslav 2 Bulíček, Miroslav 2 Ferrari, Fausto 2 Frehse, Jens 2 González Llorente, José 2 Kawohl, Bernd 2 Khabibullin, Bulat Nurmievich 2 Mocek, Tomáš 2 Nadirashvili, Nikolai S. 2 Pritsker, Igor E. 2 Saff, Edward Barry 2 Smrčka, Michael 2 Varnhorn, Werner 2 Watson, Neil A. 2 Wise, W. 2 Zhao, Zhibiao 1 Aikawa, Hiroaki 1 Arroyo, Angel 1 Bačák, Miroslav 1 Bajunaid, Ibtesam O. 1 Bao, Yong 1 Bartoletti, Silvia 1 Beck, Lisa 1 Beznea, Lucian 1 Boboc, Nicu 1 Bolfarine, Heleno 1 Bonfiglioli, Andrea 1 Brown, Russell Murray 1 Bucur, Claudia 1 Chamberland, Marc 1 Chen, Shen-Chien 1 Chen, Zhen-Qing 1 Cohen, Joel M. 1 Colonna, Flavia 1 Cosmes, Waldenia 1 Coville, Jerome 1 de Cornulier, Yves 1 Dembinski, V. 1 Di Biase, Fausto 1 Dipierro, Serena 1 Domoradzki, Stanisław 1 Dont, Miroslav 1 Dontová, Eva 1 Dostál, Petr 1 Figueiredo, C. C. 1 Freitas, Pedro 1 Fuglede, Bent 1 Gaczkowski, Michał 1 Gamboa, Fabrice 1 Gardiner, Stephen J. 1 Garg, Pratibha 1 González-Estrada, Elizabeth 1 Górka, Przemysław 1 Grigor’yan, Alexander Asaturovich 1 Groot, Alan 1 Guerra, André 1 Gustafsson, Anders 1 Haouala, Ezeddine 1 Hervé, Loïc 1 Hinz, Michael 1 Hrach, Karel 1 Huang, Tzeeming 1 Jarník, Jiří 1 Kahle, David J. 1 Kalton, Nigel John 1 Keller, Merlin 1 Khabibullin, Farkhat Bulatovich 1 Kijowski, Antoni 1 Kozubowski, Tomasz J. 1 Krantz, Steven George 1 Kraus, Michal 1 Kreuter, Marcel 1 Krutitskii, Pavel A. 1 Kumagai, Takashi 1 Kundu, Subiman 1 Kuznetsov, Nikolai Germanovich 1 Ledoux, James 1 Li, Xiaoye 1 Lima, Claudia R. O. P. ...and 58 more Authors
all top 5

#### Cited in 82 Serials

 17 Potential Analysis 14 Czechoslovak Mathematical Journal 5 Journal of Mathematical Analysis and Applications 5 Transactions of the American Mathematical Society 5 Statistics & Probability Letters 4 Israel Journal of Mathematics 4 Annali di Matematica Pura ed Applicata. Serie Quarta 4 Archiv der Mathematik 4 Mathematische Annalen 3 Journal of Approximation Theory 3 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 3 Proceedings of the American Mathematical Society 3 Applications of Mathematics 3 Journal of Mathematical Sciences (New York) 3 Bulletin des Sciences Mathématiques 2 Bulletin of the Australian Mathematical Society 2 International Journal of Theoretical Physics 2 Journal d’Analyse Mathématique 2 Mathematical Methods in the Applied Sciences 2 Arkiv för Matematik 2 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 2 Journal of Econometrics 2 Kybernetika 2 The Journal of Geometric Analysis 2 Stochastic Processes and their Applications 2 Expositiones Mathematicae 2 St. Petersburg Mathematical Journal 2 Positivity 2 Lobachevskii Journal of Mathematics 2 Communications on Pure and Applied Analysis 1 Applicable Analysis 1 Archive for History of Exact Sciences 1 Ukrainian Mathematical Journal 1 Advances in Mathematics 1 Annales de l’Institut Fourier 1 Functional Analysis and its Applications 1 Illinois Journal of Mathematics 1 Inventiones Mathematicae 1 Journal of Differential Equations 1 Journal of the London Mathematical Society. Second Series 1 Journal of Statistical Planning and Inference 1 Manuscripta Mathematica 1 Mathematica Scandinavica 1 Mathematische Zeitschrift 1 Proceedings of the London Mathematical Society. Third Series 1 Results in Mathematics 1 SIAM Journal on Control and Optimization 1 Siberian Mathematical Journal 1 Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 1 Topology and its Applications 1 Advances in Applied Mathematics 1 Ergodic Theory and Dynamical Systems 1 Revue de Statistique Appliquée 1 Acta Mathematica Hungarica 1 Acta Applicandae Mathematicae 1 Discrete & Computational Geometry 1 Journal of Theoretical Probability 1 Historia Mathematica 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 Journal of Statistical Computation and Simulation 1 SIAM Journal on Optimization 1 Russian Mathematics 1 Calculus of Variations and Partial Differential Equations 1 Bernoulli 1 Izvestiya: Mathematics 1 Annales Academiae Scientiarum Fennicae. Mathematica 1 Revista Matemática Complutense 1 Journal of Applied Statistics 1 Journal of the European Mathematical Society (JEMS) 1 Acta Mathematica Sinica. English Series 1 Stochastic Environmental Research and Risk Assessment 1 Methodology and Computing in Applied Probability 1 Journal of the Australian Mathematical Society 1 Advanced Nonlinear Studies 1 Comptes Rendus. Mathématique. Académie des Sciences, Paris 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 1 Electronic Journal of Statistics 1 Banach Journal of Mathematical Analysis 1 Advances in Calculus of Variations 1 Analysis & PDE 1 Antiquitates Mathematicae 1 Communications in Mathematics and Statistics
all top 5

#### Cited in 35 Fields

 82 Potential theory (31-XX) 49 Partial differential equations (35-XX) 34 Functional analysis (46-XX) 28 Probability theory and stochastic processes (60-XX) 15 Statistics (62-XX) 10 Functions of a complex variable (30-XX) 7 Real functions (26-XX) 6 Measure and integration (28-XX) 6 Harmonic analysis on Euclidean spaces (42-XX) 6 Operator theory (47-XX) 6 Numerical analysis (65-XX) 5 Abstract harmonic analysis (43-XX) 5 Quantum theory (81-XX) 4 History and biography (01-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 3 Topological groups, Lie groups (22-XX) 3 Several complex variables and analytic spaces (32-XX) 3 Integral equations (45-XX) 3 Convex and discrete geometry (52-XX) 3 General topology (54-XX) 3 Global analysis, analysis on manifolds (58-XX) 3 Fluid mechanics (76-XX) 2 Combinatorics (05-XX) 2 Difference and functional equations (39-XX) 2 Integral transforms, operational calculus (44-XX) 2 Geometry (51-XX) 2 Differential geometry (53-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 General and overarching topics; collections (00-XX) 1 Nonassociative rings and algebras (17-XX) 1 Special functions (33-XX) 1 Ordinary differential equations (34-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Approximations and expansions (41-XX) 1 Mechanics of deformable solids (74-XX)

#### Wikidata Timeline

The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata.


text:
## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$1$
Since any expression (except $0$) raised to $0$ is equal to $1$, then the given expression, $(8+5)^0$, simplifies to \begin{array}{l}\require{cancel} 13^0 \\\\= 1 .\end{array}


text:
## Saturday, 30 November 2013

### November leaves

November has been so mild that many of the trees still have their leaves.

 apple tree
 maple
That's not to say there aren't many leaves on the ground, and in the water.

 leaves in the rill

 leaves surrounded

It's not all leaves.

 this tiny mushroom in the path has caught a thread

## Friday, 29 November 2013

### Venus

Venus is very bright tonight.

 16:52 GMT, looking south west; unretouched phone photo
Once I had checked it wasn’t an aeroplane, I was sure it was Venus.  But I thought I’d check with Google Sky map; it’s always nice to have an excuse to play with that app.  I fired it up, and pointed my phone at the bright light.

No, I don’t think so.  And not just because I knew I wasn’t looking south…

I waited a few seconds, until the orientation sensor kicked in properly, and the image slewed round to show:

That’s better!

## Wednesday, 27 November 2013

### with just a hint mackerel

 16:10 GMT, looking west

## Monday, 25 November 2013

### graphics can have a grammar, too

My random walk through web space led me to the Python library bokeh.  This led me to Leland Wilkinson’s The Grammar of Graphics, which I will almost certainly be buying shortly, after skimming through the Google Books version.

Meanwhile, I found this very nice partial review of it, which has links to other articles and books (that I will almost certainly be buying shortly...)  It also has some excellent acerbic comments on the Excel Chart Wizard!

 This is NOT the approach that Wilkinson takes.

For all my social networking posts, see my Google+ page

## Sunday, 24 November 2013

### The Five(ish) Doctors Reboot

And for those who wanted more Original Doctors in their anniversary celebrations, try this!  (And wait until the end of the first credits, too.)

For all my social networking posts, see my Google+ page

### Coal Hill School

I got that the beginning of The Day of the Doctor was set in the actual school that Susan went to in the very first episode, and that was used for a nod back to 1963 in the 1988 story Remembrance of the Daleks, but I missed this tidbit:
We got just long enough a look at the sign with the school’s name for me to notice that the Governor of its Board of Directors is one “I. Chesterton”.
Sweet.

For all my social networking posts, see my Google+ page

## Saturday, 23 November 2013

### Spoilers!

Wow!

I had been carefully managing my expectations, because The Day of the Doctor couldn’t possibly live up to the hype, excitement, and full weight of the 50 year history, could it?

Yes it could.

 the 11 Doctors – but actually it should be 12!

We now have a continuous timeline of regenerations, filling in the gap between McGann and Ecclestone.  Which leads to the question: what is the official numbering now?  (That certainly promises oodles of fun for pedants in future pub quizzes!)  And the problem to do with the 13th regeneration has been acknowledged, so we have that to look forward to.

But most important of all, it is a great story, playing on the existence of time travel and time paradoxes, examining the great Time War and the Doctor’s dreadful role in it.  It is a marvelous episode, full of fun, pathos, wit, momentous moral decisions, and gorgeous nods to the 50 year history, from the cheeky opening shot in Totter’s Lane to the epic final battle around Gallifrey.  Tennant and Smith work excellently together, playing off each other.  In a clever twist it is these somewhat childish later Doctors who are the living with the memory of the terrible deed they have done, while the older, more adult Doctor (Hurt) has not yet committed the act that will scar the Doctor throughout his future regenerations.

 Three Doctors – the Daleks don’t stand a chance!

Who would have thought, 50 years ago today, that the future would hold something like this?

So, I wonder: what the diamond anniversary will hold?

### found it!

After trawling through more newsagents than I would like to admit, we now have the full set of Radio Times covers:

 Matt Smith found at last

### November, 1963 (Oh, What A Night)

Oh, what a night
Late November, back in ’63
Was a very special time for me
As I remember, what a fright

Oh, what a night
You know, we never got know his name
But life was never gonna be the same
What a Doctor, what a sight

Oh, I
I got a funny feeling when he walked through the door
Oh, my
That TARDIS, inside was so much more

Oh, what a sight
Materialising: mesmerizing me
Who was everything I dreamed it’d be
So SFnal, what a night

And I felt a rush when the TARDIS sounded thunder
Spinning its light around and sparking my sensawunda
Oh, what a night

Oh, I
I got a funny feeling when he walked through the door
Oh, my
That TARDIS, inside was so much more

Oh, what a night
Why’d it take so long to do what’s right?
Beeb was wrong, but now it’s seen the light
What a Doctor, what a night

And I felt a rush when the TARDIS sounded thunder
Spinning its light around and sparking my sensawunda
Oh, what a night (doo de-do, doo de-do, doo de-do, dee di-di)

### golden anniversary anticipation

We have a small collection of small daleks

standing at the foot of our smaller collection of full sized daleks.

We don’t collect only daleks, of course.  To celebrate the 50th anniversary, this week’s issue of the Radio Times has 12 different covers.

 We have 11 of the 12 different covers this week.  Matt Smith had sold out.

These will be added to our collection of other Who-covered Radio Times.

 Row 2 has 4 covers from the 40th anniversary, which form a montage.

And now, we have to wait until this evening for The Day of the Doctor, which we will be watching at home on the TV in 2D, the way it is meant to be seen!  We have, of course, already watched the webisode Night of the Doctor, which provides an crucial insight into the timeline, and the number line.

## Friday, 22 November 2013

### prediction shouldn't be this difficult

I have previously noted the inaccuracy of some computer predictions.  Tonight I observed how it also applies to cars.

As I started my trip, the GPS said the journey would take 2 hours 20 minutes, and the car instrument panel claimed I had 205 miles until an empty tank.

50 miles, and one hour, later, the GPS said the remaining journey would take 2 hours 5 minutes; more bizarrely still, the car instrument panel claimed I now had 215 miles until an empty tank.

So, it seems that the car thought I had taken 15 minutes to travel 10 miles backwards?

## Thursday, 21 November 2013

### 600 and falling

 Terrence Tao
Here’s a very nice article on that latest on closing the gap in the Twin Primes Conjecture.

tl;dr: It’s down to 600.

For all my social networking posts, see my Google+ page

### this is what happens when you give Computer Scientists access to piped chocolate

We had a Children in Need quiz night at work yesterday.  One round was to decorate a cake. Our team (The Flying Robots) felt that a fractal, with its infinite chocolate complexity, should be a sure winner.

 Sierpinski cake
The judges, for some unfathomable reason, preferred another:

 double decker artistry
Okay, maybe we weren’t taking it that seriously…

Thanks to Richard Hawkins, Gary Plumbridge, Simon Poulding, Ian Gray, Ed Powely and Sarah Christmas for some fiendish questions, as well as lovely cake!

## Tuesday, 19 November 2013

### autumn trees

Walking over to a meeting, I spotted the sun low on the magnificent tree in front of the library.

Magnificent autumn colour (and it looker oranger in real life than in the pic).  Then a few steps further on I saw another tree catching the light:

It was also brighter in real life than here.

Just, wow!

## Monday, 18 November 2013

### python's lunchbox

A love letter to Python.  The last footnote mentions bokeh, a python library supporting the Grammar of Graphics.  I know what I'm going to be playing with this weekend!

For all my social networking posts, see my Google+ page

## Sunday, 17 November 2013

### representations, permutations, visualisations

One of the things I’m interested in is evolutionary algorithms (EAs), and how to make them better.  An EA takes a population of “genomes”, “mutates” (changes a little) and selects (based on “fitness”), and mutates and selects, and ... until a suitably fit answer is found.

A recent advance has been the introduction of “evo-devo” algorithms.  (I’m putting all this biological terminology in scare quotes, because by the time the relevant process has been translated into a computer algorithm, it is so far removed from its biological inspiration as to make a biologist wince, or even exclaim in outrage.)  Evo-devo puts a distance between the genome (the representation that gets mutated) and the “phenotype” (the representation that gets selected based on fitness).  This can help the algorithm’s performance, by allowing simple easily mutatable genomes develop into complex structured phenotypes.

A colleague of mine at York, Julian Miller, is the inventor of such an algorithm, Cartesian Genetic Programming (CGP).  The genome is a string of numbers (numbers are easy to mutate a little bit).  The string is then interpreted as a network phenotype.  The network itself has inputs and outputs, so is a form of program.

Now, let’s consider permutations.  A permutation of the numbers 1 to N is these numbers in some specific order.  So the permutations of 1 to 3 are: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).  A string of length N has N! (N factorial) permutations.  N! grows very fast; while 5! = 120, 10! = 3,628,800, and 100! > 10157.

Permutations are common in computer science.  One classic use is in the Travelling Salesman Problem: given a bunch of N cities, find the shortest path through all of them.  That is, find the permutation of 1 to N that gives the shortest path.  Given there are N! such permutations, clearly we don’t want to try them all.  Although exact algorithms that are essentially more efficient than trying all possibilities aren’t known (and it is strongly suspected that there aren’t any), there are algorithms that come up with very good approximate (nearly shortest path) answers most of the time.  EAs are one such class of algorithms: breed for fitter (shorter) paths.

Permutations as genomes are a bit tricky, though.  A permutation has structure: it must contain all the numbers from 1 to N, and each only once.  So you can’t mutate a single entry: you have to swap two entries, or do something else that maintains the permutation structure. “Crossover” is even harder: how do you take half of one permutation, half of another, and combine them into a valid permutation?  There are various techniques, but they are not very pretty.

Using an evo-devo approach to generate a permutation seems even harder: how do you ensure that your developed system is a valid permutation?  So, for example, with CGP we can have a list of outputs, but how do we ensure that this list is a valid permutation?  (Having a single output that is already a permutation merely moves the problem back inside the network somewhere.)

We need a further representation and development step that is guaranteed to produce a permutation.  Rather than try to get the network to produce a permutation immediately, let’s break it down into two steps: the network produces a list of numbers, then that list has to go through a further interpretation step to form a permutation.  Julian came up with an idea of how to do this: given a list of (say) real numbers (easy to produce with CGP), just sort them into ascending order.  The correspondingly sorted list of the indexes gives the required permutation.  Voilà!

 A string of numbers is interpreted as a network, which outputs a real vector, which when sorted yields a permutation

What is happening here is easy to visualise using a technique called parallel coordinates.  A list of N real numbers can be thought of as a vector in N-D space.  But N-D space is hard to visualise if N > 3.  (I find it pretty hard to visualise even when N = 3.)

 It’s hard to visualise a lot of dimensions this way

Parallel coordinates do what it says in the name: instead of drawing the N dimensions orthogonal to each other (rapidly running out of ways to do this in our 3D physical space), draw them parallel to each other.  It’s easy to draw lots of parallel lines.  Now plot the N-D point (x1,x2,...xN) as follows: plot the point x1 on axis 1, the point x2 on axis 2, and so on, then joint these points together with a line.  The line in the parallel coordinate plot represents the point in N-D space.

 parallel coordinates view of a single N-D point

We can use these parallel coordinated to visualise how a vector of real numbers can represent a permutation by its components being sorted into ascending order.

 (top) a vector of 20 real numbers, a 20-D point, drawn in parallel coordinates; (bottom) the same vector, with the parallel axes ordered so that the components are in increasing order: the sorted axis indexes are the permutation represented by the N-D point.

The Python/numpy code that generated these plots is:
N = 20
P = range(N) # indexes
V = rand(N)  # random vector

# plot unsorted vector
for dim in range(N):
ax1.plot([dim, dim], [0, 1], '0.5', linewidth=0.25)
ax1.text(dim, -0.2, str(P[dim]), ha='center', fontsize=32)
ax1.plot(range(N), V, '.k', markersize=20)
ax1.plot(range(N), V, 'k')

# sort, and plot sorted vector
P = argsort(V) # sort indexes
V = sort(V)    # sort vector (for plotting)
for dim in range(N):
ax2.plot([dim, dim], [0, 1], '0.5', linewidth=0.25)
ax2.text(dim, -0.2, str(P[dim]), ha='center', fontsize=32)
ax2.plot(range(N), V, '.k', markersize=20)
ax2.plot(range(N), V, 'k')
Note that the code that generates the permutation is the single line P = argsort(V): the rest is just plotting code.

Here I started from a random vector, rather than the non-random output of some CGP network.  Sorting a random vector is one way to construct a random permutation, but as far as Julian and I can tell from the literature, this CGP use for representing evolved, non-random permutations isn’t standard.  Julian has been using it for several years in his module on evolutionary algorithms, and will be publishing a paper on some results next year.

## Friday, 15 November 2013

### questioning scientists

I don’t watch Question Time, but signed this petition anyway. If they respond positively, I might even start watching!

 Question Time representation

For all my social networking posts, see my Google+ page

## Thursday, 14 November 2013

### remote interaction

want

(via Hiroki Sayama)

For all my social networking posts, see my Google+ page

## Wednesday, 13 November 2013

### afternoon sunset II

a little earlier than yesterday, and more spectacular

 16:18 GMT, looking west

## Tuesday, 12 November 2013

### afternoon sunset

The view at 4:30pm, and it’s still over a month to the shortest day!

 16:30 GMT, looking west

## Saturday, 9 November 2013

### the old and the new

I’m just back from a 2 day residential Theo Murphy scientific workshop held in the Royal Society’s Kavli Centre at Chicheley Hall.  It’s my first visit there, and I can certainly recommend it as a marvellous venue for a workshop: great facilities, marvelous food, and friendly, efficient staff.  The science was great fun: I learned lots of new things, discovered links between seemingly diverse areas, and had interesting discussions over food and coffee.  I’m buzzing with ideas, which is the whole point!

I did the usual “photograph from my bedroom window” thing, which had a somewhat different from usual view:

 first day of the workshop, view due east, into the rising sun

Oh, and then I took a photo from the other window in my bedroom:

 first day of the workshop, view due south
Blissful.  I found an amusing view from the window half way down the main stairs:

A lovely formal garden, with lawns, paths, clipped bushes and trees, and, just visible at the vanishing point of the path … a wind farm!  The old and the new collide.

### sequestering carbon, several books at a time XII

A rather small haul:

## Monday, 4 November 2013

### a deeply imbedded myth

How many cell types are there?  Another myth laid bare...

(I also recommend the referenced Fox Terrier paper.)

 There is no data supporting this kind of chart.
this number and imaginary trend in cell type complexity are derived entirely from an otherwise obscure and rarely cited 60 year old review paper that contained no original data on the problem; the values are all guesswork, estimates from the number of cell types listed in histology textbooks.

For all my social networking posts, see my Google+ page

## Sunday, 3 November 2013

### what the British say -- what the British mean

So true (the yellow column, at least -- is the pink column really what you think we mean?!)

For all my social networking posts, see my Google+ page

## Saturday, 2 November 2013

### 50 years ago

From an editorial in this week’s edition of Nature:
November sees perhaps the most significant golden anniversary to have been celebrated anywhere or anywhen in the entirety of time and space. Yes — it will have been 50 years since the broadcast of the first episode of Doctor Who.
... and I remember the first episode ...

(via Tim Taylor)

For all my social networking posts, see my Google+ page

## Friday, 1 November 2013

### ubiquitous destruction

I just saw the following on BoingBoing:

 The “Destruct Room” in Jack Kirby’s comic book OMAC (1974) was a place where stressed-out people could act on urges to smash things.
It reminded me of:

 A panel from “The Gabriel Set-Up”, the third story in the Modesty Blaise comic strip, by Peter O’Donnell and artist Jim Holdaway (1964)
Clearly the urge to smash is universal.


text:
# Show that $\ker(\phi)$ is finitely generated

$$\DeclareMathOperator\Ker{Ker}$$Let $$\phi : M \to F$$ be a surjective homomorphism of a finitely generated module $$M$$ onto a free module $$F$$. Show that $$\Ker(\phi)$$ is finitely generated.

My attempt :

I considered the Short Exact Sequence :$$0 \to \Ker(\phi) \to M \to F \to 0.$$ The map from $$\Ker(\phi) \to M$$ is the inclusion and the map from $$M\to F$$ is $$\phi$$. Since, $$F$$ is free, $$\exists \psi : F \to M$$ (R-linear map) such that $$\phi \circ \psi = id_F$$ , in other words , the above Short Exact Sequence splits and hence,$$M \cong\Ker(\phi) \oplus F$$ . Let $$j:M \to \Ker(\phi) \oplus F$$ be the isomorphism.

Given any $$k \in \Ker(\phi)$$ and given $$m \in M$$ we get a unique, $$f \in F$$ such that $$j(m)=k+f$$.

Now since, $$M$$ is finitely generated $$\exists m_1,\dots,m_l,\lambda_1,\dots,\lambda_l$$ such that, $$m= \lambda_1 m_1 + \dots + \lambda_l m_l$$ and also since , $$F$$ is free, we get a basis $$\{f_1,\dots,f_s\}$$ and thus $$f=c_1f_1 + \dots +c_sf_s$$ and hnece, we can write,$$k=\lambda_1j(m_1)+\dots+\lambda_lj(m_l)-c_1f_1-\dots-c_sf_s$$

Since , $$k\in \Ker(\phi)$$ was arbitrarily chosen we get that $$\Ker(\phi)$$ is generated by$$\{j(m_1),\dots,j(m_l),f_1,\dots,f_s\}$$ and hence in particular finitely generated.

There might be answers else where, But I want to check whether my arguments are correct before looking into others' answers. So please do not tag this as duplicate.

$$\DeclareMathOperator\Ker{Ker}$$Since $$F$$ is a free, hence projective, module, the sequence $$\{0\}\to\Ker (\phi)\to M\to F\to\{0\}$$ splits. Consequently, $$\Ker (\phi)$$ is a direct summand, hence an homomorphic image, of the finitely generated module $$M$$, thus it is finitely generated as well.
No, $$\ker\phi$$ is not generated by $$\{j(m_1),\dots,j(m_l),f_1,\dots,f_s\}$$, but just by $$\{j(m_1),\dots,j(m_l)\}$$. The former (bigger) set generates $$\ker\phi\oplus F$$.
Given any $$k\in\ker\phi$$ and given $$m\in M$$, we get a unique $$f\in F$$ such that $$j(m)=k+f$$.
You should write: given $$k\in\ker\phi$$, there exists a unique $$m\in M$$ such that $$k=j(m)$$.
Since $$M$$ is finitely generated, say by $$\{m_1,\dots,m_l\}$$, we can write $$m=\lambda_1m_1+\dots+\lambda_lm_l$$. This proves that $$\ker\phi$$ is generated by $$\{j(m_1),\dots,j(m_l)\}$$.


text:
Recover the Jianguoyun Tray Icon in Ubuntu 18.04

2018-10-17

I cannot find the Jianguoyun (Nutstore) icon in the top tray after it runs. Here is a quick fix.

The original post link: 在 Ubuntu18.04 系统顶栏不显示坚果云图标的解决办法 - 故国有明

Problem

I primarily use Jianguoyun to synchronize files with collaborators in China, since Dropbox is banned by China GFW.

There is no icon shown in the tray on the top of the screen after I run the Jianguoyun (also known as Nutstore) cloud drive. So you get trouble when you want to change the settings.

Reason

The latest Gnome discards the old version system tray. However, Jianguoyun still uses the old version. We need another extension to add the old system tray icons.

Solution

• Install gnome-tweak-tool. I believe the majority of the users have it.
• Install TopIcons Plus extension for Gnome. Once it is installed, the Jianguoyun icon pops up immediately (as the top image shows).
• Update settings in Gnome Tweaks.

Notice

If Jianguoyun updates their linux app in the future, this post will not work. I may update the post at that time.


text:
## The Annals of Probability

### A Solution to the Game of Googol

Alexander V. Gnedin

#### Abstract

For any $n > 2$ we construct an exchangeable sequence of positive continuous random variables, $X_1, \ldots, X_n$, for which, among all stopping rules, $\tau$, based on the $X$'s, $\sup_\tau P\{X_{\tau} = X_1 \vee \cdots \vee X_n\}$ is achieved by a rule based only on the relative ranks of the $X$'s.

#### Article information

Source
Ann. Probab., Volume 22, Number 3 (1994), 1588-1595.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988613

Digital Object Identifier
doi:10.1214/aop/1176988613

Mathematical Reviews number (MathSciNet)
MR1303655

Zentralblatt MATH identifier
0815.60038

JSTOR
links.jstor.org

#### Citation

Gnedin, Alexander V. A Solution to the Game of Googol. Ann. Probab. 22 (1994), no. 3, 1588--1595. doi:10.1214/aop/1176988613. https://projecteuclid.org/euclid.aop/1176988613


text:
Next: Signal Through System in Up: Fourier Transforms and Signal Previous: Vector Space and Orthogonal

## Orthogonal transform as rotation of basis vectors

The discrete Fourier transform, as one of many possible orthogonal transforms (e.g., discrete cosine transform, etc.), can be considered as a rotation of the standard basis vectors. A signal as a N-D vector is represented implicitly by the standard basis. It can be equivalently represented by any other basis, some rotated version of the standard basis. This idea can be illustrated by the following example of an N=2 dimensional vector space.

A 2-D vector space is spanned by two standard basis vectors:

A vector can be represented by these basis vectors as:

This 2-D vector space can also be spanned by another basis composed of two vectors, such as the standard basis rotated clockwise by . This rotation is represented by a rotation matrix:

Obviously this matrix is orthogonal:

The new basis vectors can then be obtained as:

The same vector can also be represented by the new basis:

The coefficients and can be obtained as the projections of onto the two basis vectors:

Now the same vector can also be expressed as:

We see that the vector can be equivalently represented by either and , or and , depending on which basis is used.

The vector can be considered as a discrete signal of samples:

In fact, this rotation is the discrete Fourier transform of signal of samples:

Conclusion:

The N-point Discrete Fourier transform is essentially a rotation of the standard basis vectors to a set of new basis vectors representing different frequencies. More generally, each of the infinitely many possible rotations of the standard basis vectors corresponds to a certain orthogonal transform, and the Fourier transform is only one of these transforms. Also as an orthogonal transform is simply a rotation of the basis of the vector space, it does not change the norm (length) of a vector, i.e., the energy contained in a signal is conserved before and after the transform, this is the Parseval's identity.

Next: Signal Through System in Up: Fourier Transforms and Signal Previous: Vector Space and Orthogonal
Ruye Wang 2010-01-16


text:
# Calculate OLS estimator manually in R

This post shows how to manually construct the OLS estimator in R (see this post for the exact mathematical derivation of the OLS estimator). The code will go through each single step of the calculation and estimate the coefficients, standard errors and p-values.  In case you are interested the coding an OLS function rather than in the step wise calculation of the estimation itself I recommend you to have a look at this post

For demonstration purpose, we will construct a fake data set with simulated height and weight data. We will calculate the relationship between height and weight using the $lm()$ function of R. The $lm()$ function is the build-in OLS estimator of R. We will then continue to construct the OLS estimator ourselves and estimate the coefficients ($\beta$) of the relationship between height and weight and calculate the standard errors ($s.e.$) around the estimated betas. Finally, we will compare the output of $lm()$ and our manual constructed estimator and show that they are equivalent.

The following code simulates the data we are going to use

rm(list=ls())

## create artificial data set
#  set seed for reproducibility
set.seed(501)

#  height in cm
height <- rnorm(200,175,sd = 15)

#  weight in kilogram (the relationship between height
#  and weight is completely random)
weight <- height-80+1.02*(height)^0.01*
rnorm(200,0,sd = 15)*
rnorm(200,1.1,sd = 0.2)

#  join height and weight in data frame
df <- data.frame(height,weight)

We will regress height on weight, after the construction of the data set.

## use R build-in OLS estimaor (lm())
reg = lm(weight ~ height, data=df)
summary(reg)

Finally, we will construct the OLS estimator manually and compare the results to the lm() output. You will see that they are equivalent.

## build OLS estimator manually

#  define X matrix and y vector
X <- as.matrix(cbind(1,df$height)) y <- as.matrix(df$weight)

#  estimate the coeficients beta
#  beta = ((X'X)^(-1))X'y
beta <- solve(t(X)%*%X)%*%t(X)%*%y

## calculate residuals
#  res = y - beta1 - beta2*X2
res <- as.matrix(y-beta[1]-beta[2]*X[,2])

## define the number of observations (n) and the number of
#  parameters (k)
n <- nrow(df)
k <- ncol(X)

## calculate the Variance-Covariance matrix (VCV)
#  VCV = (1/(n-k))res'res(X'X)^(-1)
VCV <- 1/(n-k) * as.numeric(t(res)%*%res) * solve(t(X)%*%X)

## calculate standard errors (se) of coefficients
se <- sqrt(diag(VCV))

## calculate the p-values
p_value <- rbind(2*pt(abs(beta[1]/se[1]), df=n-k,
lower.tail= FALSE),
2*pt(abs(beta[2]/se[2]), df=n-k,
lower.tail= FALSE))

## combine all necessary information
output <- as.data.frame(cbind(c("(Intercept)","height"),
beta,se,p_value))
names(output) <- c("Coefficients:","Estimate",
"Std. Error","Pr(>|t|)")

#compare automatic (lm) and manual output
output
summary(reg)

This post showed how to compute the OLS estimator in R by are going slowly through each step of the calculation. The following post takes OLS estimation in R to the next level and wraps a function around the presented code. Ultimately, it is far more convenient to conduct analysis when using function rather than consistently going through each step of the calculation.

## 4 thoughts on “Calculate OLS estimator manually in R”

1. sam says:

# beta = ((X’X)^(-1))X’y
beta <- solve(t(X)%*%X)%*%t(X)%*%y

where is ^(-1) in the code?


text:
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.

QUESTION

# What is the lewis structure for SO_2?

Here are the steps I follow when drawing a Lewis structure.

1. Decide which is the central atom in the structure. That will normally be the least electronegative atom ("S").

2. Draw a skeleton structure in which the other atoms are single-bonded to the central atom: "O-S-O".

3. Draw a trial structure by putting electron pairs around every atom until each gets an octet. In this editor, I will have to write it as ::Ö-S(::)-Ö::

4. Count the in your trial structure (20).

5. Now count the you actually have available. "1 S + 2 O = 1×6 + 2×6 = 18". The trial structure has two extra electrons.

6. Draw a new trial structure, this time inserting one double bond for each extra pair of electrons:

"O=S-O" and "O-S=O"

7. As before, add valence electrons to give each atom an octet:

8. Calculate the formal charge on each atom.

9. We see that some of the atoms have formal charges. The “best” Lewis structure is one in which has the fewest formal charges. We can generate a structure with zero formal charges if we move a lone pair from the single-bonded "O" to make a double bond to the "S".

This gives us a third possibility:

We now have a structure in which "S" has ten valence electrons. However, that’s OK, because "S" can “expand” its octet.

We have three different structures, differing ONLY in the locations of the electrons. We say that these are resonance structures of "SO"_2.

The actual structure of "SO"_2 is a resonance hybrid of all three structures.

LEARN MORE EFFECTIVELY AND GET BETTER GRADES!


text:
# Fonts issues in xelatex

I would like to know what is the right way to define all the text font as a specific size. I use \documentclass[20pt]{article} but it won't change the font size in xelatex. And these symbols (" and ') are not Times fonts and now matter how I changed the fonts , they look the same. Here is the min working code. I would like to know what is the right way to address font selection in xelatex. Can I get like this ?

 %!TEX TS-program = xelatex

\documentclass[20pt]{article}
\usepackage{geometry}
\usepackage{mathspec}
\usepackage{lipsum}
\setmainfont{Times}
\setmathsfont(Digits,Latin,Greek){Times}
\usepackage{titling}
\usepackage{titlesec}
\usepackage{setspace}
\doublespacing
\titleformat{\section}{\singlespacing\fontsize{20}{24}\bfseries\rmfamily\uppercase} {\thesection.}{0.5em}{}
\begin{document}
\section{"check"}
"just checking" this is a "check" and 'check'.
\end{document}

-
possible duplicate of Fonts larger than \Huge? – DG' Feb 25 '14 at 8:29
To get the proper quotes, add the option Mapping=tex-text to your \setmainfont command. Opening quotes are typed with  and ; closing quotes are typed with ' and ''. Don't use ", since it has incorrect spacing. – ChrisS Feb 25 '14 at 8:54

In case if you are tied to article document class, here are the two options:

Option-1

You can use scrextend package, that extends the facilities (like font sizes) of KOMA-script classes to other classes:

%!TEX TS-program = xelatex

\documentclass{article}
\usepackage{scrextend}         %%% this line
\changefontsizes[24pt]{20pt}   %%% and this line
\usepackage{geometry}
\usepackage{mathspec}
\usepackage{lipsum}
\setmainfont[Ligatures=TeX]{Times New Roman}
\setmathsfont(Digits,Latin,Greek){Times New Roman}
\usepackage{titling}
\usepackage{titlesec}
\usepackage{setspace}
\doublespacing
\titleformat{\section}{\singlespacing\fontsize{20}{24}\bfseries\rmfamily\uppercase} {\thesection.}{0.5em}{}
\begin{document}
\section{check''}
just checking'' this is a check'' and check'.
\end{document}

Option-2

You can use extsizes package, which is a package version of extarticle mentioned by Karl Koeller in his answer.

%!TEX TS-program = xelatex

\documentclass{article}
%\usepackage{scrextend}
%\changefontsizes[24pt]{20pt}
\usepackage[20pt]{extsizes} %%% this line
\usepackage{geometry}
\usepackage{mathspec}
\usepackage{lipsum}
\setmainfont[Ligatures=TeX]{Times New Roman}
\setmathsfont(Digits,Latin,Greek){Times New Roman}
\usepackage{titling}
\usepackage{titlesec}
\usepackage{setspace}
\doublespacing
\titleformat{\section}{\singlespacing\fontsize{20}{24}\bfseries\rmfamily\uppercase} {\thesection.}{0.5em}{}
\begin{document}
\section{check''}
just checking'' this is a check'' and check'.
\end{document}

-
– Johannes_B Jan 17 '15 at 13:22

The article class only supports 10pt, 11pt, and 12pt.

You can use the extarticle (extended article) class which supports 20pt.

In the following MWE I've changed "Times" to "Times New Roman" and I get the correct quotes using Ligatures=TeX option, but you have to use the standard LaTeX way for double quotes.

%!TEX TS-program = xelatex

\documentclass[20pt]{extarticle}
\usepackage{geometry}
\usepackage{mathspec}
\usepackage{lipsum}
\setmainfont[Ligatures=TeX]{Times New Roman}
\setmathsfont(Digits,Latin,Greek){Times New Roman}
\usepackage{titling}
\usepackage{titlesec}
\usepackage{setspace}
\doublespacing
\titleformat{\section}{\singlespacing\fontsize{20}{24}\bfseries\rmfamily\uppercase} {\thesection.}{0.5em}{}
\begin{document}
\section{check''}
just checking'' this is a check'' and check'.
\end{document}

Output

If you want "..." to behave this way, you can add these two lines in your preamble:

\usepackage{csquotes}
\MakeOuterQuote{"}

-
I thought if I use xelatex , I would get the identical fonts I can get in other places like the image I just updated. Is it not possible even though I use the same fonts? Thanks, – Aung Feb 25 '14 at 8:42
@Aung I've updated the answer. – karlkoeller Feb 25 '14 at 9:07

use

\documentclass[fontsize=20pt]{scrartcl}
`
-


text:
# Quark Matter 2018

13-19 May 2018
Venice, Italy
Europe/Zurich timezone
The organisers warmly thank all participants for such a lively QM2018! See you in China in 2019!

15 May 2018, 15:20
20m
Sala Volpi, 1st Floor (Palazzo del Casinò)

### Sala Volpi, 1st Floor

#### Palazzo del Casinò

Parallel Talk Thermodynamics and hadron chemistry

### Speaker

Mr Jean-Bernard Rose (Frankfurt Institute for Advanced Studies)

### Description

Previous calculations of the shear viscosity to entropy density ratio in the hadron gas have failed to reach a consensus, with $\eta/s$ predictions differing by almost an order of magnitude. This work addresses and solves this discrepancy by providing an independent extraction of this coefficient using the newly-developed SMASH (Simulating Many Accelerated Strongly interacting Hadrons) transport code and the Green-Kubo formalism. We compare the results from SMASH with numerical solutions of the Boltzmann equation for various systems using the Chapman-Enskog expansion as well as previous results in the literature. Substantial deviations of the coefficient are found between transport approaches mainly based on resonance propagation with finite lifetime (such as SMASH) and other (semi-analytical) approaches with energy-dependent cross-sections, where interactions do not introduce a timescale other than the inverse scattering rate. Our conclusion is that long-lived resonances strongly affect the transport properties of the system, resulting in significant differences in η/s with respect to other approaches where binary collisions dominate. We argue that the relaxation time of the system —which characterizes the shear viscosity— is determined by the interplay between the mean-free time and the lifetime of resonances. We finally show how an artificial shortening of the resonance lifetimes or the addition of a background elastic cross section nicely interpolate between the two discrepant results. To turn this around, we finally note that the temperature dependence of $\eta/s$ can be used to constrain the properties of the hadron gas.

Content type Theory Presenter name already specified

### Primary author

Mr Jean-Bernard Rose (Frankfurt Institute for Advanced Studies)

### Co-authors

Juan M Torres-Rincon (Stony Brook University) Anna Schäfer (Frankfurt Institute for Advanced Studies (FIAS))


text:
# Deriving Dark Matter; specifically looking for a table of stellar speed versus distance from center of galaxy to derive dark matter

I'm trying to find some data that I can use to derive dark matter (in a loosey-goosey sort of way, I won't be too rigorous). I'm helping a friend out with a final project for his astronomy course and I'm tasked with doing the maths. Any idea where I can find a table of various stars distance from the center of our galaxy and their speed (how fast they are going around the center of the galaxy)? This does not need to be comprehensive but would ideally contain several data sources from various representative radii. Also some sort of formatted text would be SUPER cool so I could just tell my computer to do the calculations!

I don't know of any sources for the data you're looking for, however, I did something like this a year ago in one of my classes. An option would be (keeping in mind that you don't want to be too rigorous) to get an image of a good enough galaxy, i.e a fits file released to the public like M83. Under the assumption that wherever there's light there's mass (mass = $kL$, where $k$ is a constant and $L$ luminosity), you can equate forces and solve for velocity. This would get you values for "expected virtual velocities" at each radii, so you could plot the data and get a curve with a certain shape. What you expect to find is that as you increase radii, the values for velocity drop under the initial assumption. Then you go and compare with different and more rigorous measurements, like this one (page 18). Here you notice that the curve doesn't quite agree with the shape of the curve you got from the image, since it peaks and stays approximately constant (as opposed to dropping). Why is that? Maybe there's something that light can't show you. Note: as I said, this isn't rigorous. I've skipped conversion from counts per pixel, relationship between flux and luminosity and some other things. I've also assumed that you can work with something like astropy to process the data. Hope it helps


text:
# Homeomorphism between connected sums

#### ModusPonens

##### Well-known member
Hello.

I'd like to know how to solve the following problem: [TEX]\mathbb{R}P^2[/TEX]#[TEX]T^2=\mathbb{R}P^2[/TEX]#[TEX]K^2[/TEX] , where [TEX]\mathbb{R}P^2[/TEX] is the real projective plane, [TEX]T^2[/TEX] is the torus and [TEX]K^2[/TEX] is the Klein bottle.

#### ModusPonens

##### Well-known member
I think I did not explain myself well when posting this.

The problem is to prove the equality. But no equations are necessary. For example, in a previous exercise it's asked to prove that $S^2$ is the identity in the operation #. It sufices to say that a sphere without a part that is homeomorphic to an open disk $D^2$ is homeomorphic to a closed disk.

If that's not the problem, could you tell me if this problem is a hard one for a 3rd year student of mathematics?


text:
Background

I attempted the Oxford Math Admission Test (MAT) in Nov 4th, 2020 and got 82 (did not get interview invite unfortunately), now I am preparing for the Cambridge math admission test (STEP) just for extra challenge.

In this blog I would give some reflections on STEP and share my personal experience, the MAT blog would be posted later. Hope this blog would help you :)

Introduction

STEP takes place in June every year, aiming at shortlisting the most talented candidates during the university application process. So unlike other university entrance exams like Alevel, HKDSE etc., STEP problems are much more difficult and challenging.

There are three papers in the examination, namely STEP1, STEP2 and STEP3, ranked according to difficulty (means 1 is the easiest and 3 is the most difficult). In recent years STEP1 is cancelled and two other exams still run as normal.

As I attempt STEP2 this year, I would mainly talk about STEP2 in this blog.

The syllabus of STEP2 can be found in this website: STEP 2021 Specifications. In general the pure math section of the exam would cover Calculus, Matrices, Polynomials, Number theory and many other topics.

My Preparation

I finished my last DSE exam - Math Extended Part 2 (Calculus and Algebra) in May 17th, and STEP2 would be in June 14th, so there was not much time left for preparing the exam.

Since I studied under the HKDSE curriculum, some content in STEP’s curriculum is not covered. Therefore, I bought Alevel Further Math’s textbook and started self-study. I learnt topics like relationship between roots in polynomial, linear transformation, and first-order differential equation on my own.

After that, I began practising real STEP problems by year, after several days I’ve done five past papers (2016 ~ 2020). Then I started to do the exercises provided in the book Advanced Problems in mathamatics. Most of the problems there are selected from STEP1 questions so they are not very difficult, but by doing that you could be familar with various topics, and could employ relevant knowledge to solve the problems in that topic during the actual exam.

In the last several days, I began doing STEP problems by topic, I classified topics that I has the most confidence, and selected relevant problems on STEP database for practising purposes.

Problem Characteristics

In one STEP problem, there are always many subproblems, starting from trivial ones to challenging ones, and guiding you through the whole question by giving hints. So the problems would not be very difficult if you can notice them, it is highly possible that you could use the results of previous subproblems to help solve the current one. Therefore, when you are attempting the question, make sure that you check the previous subproblems and see if they could give some hints and guidence.

Reflections by topic

Logic

If $P \Rightarrow Q$, then Q is necessary condition of P, P is sufficient condition of Q. If $P \Leftrightarrow Q$, then P is necessary and sufficient condition of Q.

Calculus

The calculus questions in STEP mainly test basic calculus manipulation skills such as integration by substitution, by parts etc. And sometimes would include reduction formula as well.

For by substitution questions, the first subproblem would always tell you what to substitute, for example:

By using the substitution $u = 1/x$, show that for $b > 0$:

$\int_{1/b}^b{\frac{x \ln x}{(a^2+x^2)(a^2x^2+1)}}\, dx = 0$

( STEP2 2014 Question 4 )

And in the last subproblem would ask you to choose suitable substitution to prove or evaluate something.

By using a substituion of the form $u = k/x$, for suitable k, show that:

$\int_{0}^{\infty}{\frac{1}{(a^2+x^2)^2}}\, dx = \frac{\pi}{4a^3}$

In such problem, you would have two hints: the first is the hint from the previous subproblems, and the second is the upper and lower limit of the integral, they give you idea about what to substitute.

Example 2 ( STEP2 1998 Q7 ):

Prove that, if

$I = \int_{0}^{1}{\frac{(1+x^2)^k}{(1+x)^{k+1}}}\, dx$

Then:

$I = \int_{0}^{\frac{1}{4}\pi}{\frac{d\theta}{[\sqrt{2} \cos \theta \cos (\frac{1}{4}\pi - \theta)]^{k+1}}} \, dx$

Show further that:

$I = 2\int_{0}^{\frac{1}{8}\pi}{\frac{d\theta}{[\sqrt2 \cos \theta \cos (\frac{1}{4}\pi - \theta)]^{k+1}}} \, dx = 2\int_{0}^{\sqrt{2} - 1}{\frac{(1+x^2)^k}{(1+x)^{k+1}}}\, dx$

For integration by parts problem, a lot of calculations are required so make sure that you do not make careless mistakes, STEP2 1994 Question 2 is one of the very challenging questions. Lower and upper limit of the integral is still very useful in the problem, 2003 STEP2 Question 7 is a case in point:

Show that if $n > 0$ then:

$\int_{e^{1/n}}^{\infty}{\frac{\ln x}{x^{n+1}}} \, dx = \frac{2}{n^2 e}$

From the problem we could discover that the lower limit of the integral is $e^{1/n}$, but the result contains $\frac{1}{e}$, that means the result of the integral should contain $x^{-n}$, it could only be obtained by integrating $x^{-(n+1)}$, therefore we could do the following procedure:

$\int_{e^{1/n}}^{\infty}{\frac{\ln x}{x^{n+1}}} \, dx = \int_{e^{1/n}}^{\infty}{\ln x \frac{d(-\frac{1}{n}x^{-n})}{dx}} \, dx = ... = \frac{2}{n^2 e}$

Trapezium rule is in syllabus too, STEP2 1987 Question 7 is worth attempting, which tests approximation of the integral.

Sometimes the problem needs to evaluate integral which upper or lower limit contains infinity, for example:

$\int_{0}^{\infty} x^n e^{-x}\, dx = [-x^n e^{-x}]^{\infty}_{0} + n\int_{0}^{\infty}{x^{n-1} e^{-x}} \, dx$

For $-x^n e^{-x}$, it equals to $\frac{-x^n}{e^x}$,as $e^x$ grows much faster than $x^n$, when $x \to \infty$, it approaches zero.

Matrices

Linear transformation is an important concept in STEP, including rotation, shearing, reflection etc.

STEP2 1998 Question 5, STEP2 1993 Question 10, STEP2 1989 Question 7 are good practice materials.

Besides, STEP problems may test matrices in an algebaric way, for example:

If M is a $2\times 2$ matrix, prove that:

$\mathrm{Tr}(M^2) = \mathrm{Tr}(M)^2 - 2\mathrm{Det}(M)$


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# Convergence of definite integral

I have to find out the convergence of the next integral: $$\int^{\pi/2}_0{\frac{\ln(\sin(x))}{\sqrt{x}}}dx$$ Any help? Thanks

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I've already seen this one today...but I'm very bad at finding duplicates. –  1015 Feb 24 '13 at 23:00

The only problem is at $0$.

We have, as $x$ approaches $0$, $$\sin x\sim x$$ so $$\frac{\log(\sin x)}{\sqrt{x}}\sim\frac{\log x}{\sqrt{x}}$$ and the integral of the latter converges at $0$.

If you did not know that, compare for instance with $\frac{1}{x^{2/3}}$. Since $$x^{2/3}\frac{\log x}{\sqrt{x}}\longrightarrow 0$$ we have $$\lvert \frac{\log x}{\sqrt{x}}\rvert\leq \frac{C}{x^{2/3}}.$$ Now it is easily seen that the integral of $1/x^{2/3}$ converges at $0$.

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true, but not a trivial statement. –  Ron Gordon Feb 24 '13 at 23:09
@rlgordonma Not that difficult either. Like with Bertrand series, the behavior of $\int_0 x^\alpha(\log x)^\beta dx$ is standard. –  1015 Feb 24 '13 at 23:10
To you, sure; to the OP, who knows. –  Ron Gordon Feb 24 '13 at 23:12
@rlgordonma Sure. But we can always add details, no? By the way, I did. –  1015 Feb 24 '13 at 23:12
And good details they are! Seriously, I wasn't picking on you, but it was that you labeled something as self-evident which I felt it took pains to show. –  Ron Gordon Feb 24 '13 at 23:25

If you integrate by parts, then you find $$\int_0^{\pi/2}dx\,\frac{\ln\sin x}{\sqrt{x}}=2\sqrt{x}\ln\sin x\Bigg|_0^{\pi/2}+2\int_0^{\pi/2}dx\,\sqrt{x}\frac{\cos x}{\sin x}.$$ It is fairly clear at this point that the integral converges because $\sqrt{x}/\sin x\sim x^{-1/2}$ at $x=0$.

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The tricky part of the integral is near $x=0$. There, note that $\sin{x} \sim x$, and consider

$$\int dx \frac{\log{x}}{\sqrt{x}}$$

Substitute $x=u^2$, $dx=2 u du$ and this integral is equal to

$$2 \int du u \frac{1}{u} \log{u^2} = 4\int du \log{u} = 4 (u \log{u}-u) = 2 (\sqrt{x} \log{x} - 2 \sqrt{x})$$

Thus, near $x=0$, the singularity is integrable, and the integral converges.

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complex geometry

# Contents

## Idea

### General

Generally, a theta function ($\theta$-function, $\Theta$-function) is a holomorphic section of a (principally polarizing) holomorphic line bundle over a complex torus / abelian variety. (e.g. Polishchuk 03, section 17) and in particular over a Jacobian variety (Beauville) such as prequantum line bundles for (abelian) gauge theory. The line bundle being principally polarizing means that its space of holomorphic sections is 1-dimensional, hence that it determines the $\theta$-function up to a global complex scale factor. Typically these line bundles themselves are Theta characteristics. Expressed in coordinates $\mathbf{z}$ on the covering $\mathbb{C}^g$ of the complex torus $\mathbb{C}^g/\mathbb{Z}^g$, a $\theta$-function appears as an actual function $\mathbf{z} \mapsto \theta(\mathbf{z})$ satisfying certain transformation properties, and this is how theta functions are considered.

Those theta functions encoding sections of line bundles on a Jacobian variety $J(\Sigma)$ of a Riemann surface $\Sigma$ (determinant line bundles, Freed 87, pages 30-31) typically vary in a controlled way with the complex structure modulus $\mathbf{\tau}$ of $\Sigma$ and are hence really functions also of this variable $(\mathbf{z},\mathbf{\tau}) \mapsto \theta(\mathbf{z}, \mathbf{\tau})$ with certain transformation properties. These are the Riemann theta functions. They are the expressions in local coordinates of the covariantly constant sections of the Hitchin connection on the moduli space of Riemann surfaces $\mathcal{M}_\Sigma$ (Hitchin 90, remark 4.12). In the special case that $\Sigma$ is complex 1-dimensional of genus $g = 1$ (hence a complex elliptic curve) then such a function $(z,\tau) \mapsto \theta(z,\tau)$ of two variables with the pertinent transformation properties is a Jacobi theta function. Notice that in their dependency not only on $\tau$ but also on $z$ these are properly called Jacobi forms. Finally notice that these line bundles on Jacobian varieties have non-abelian generalizations to line bundles on moduli stacks of vector bundles of rank higher than one, whose sections may then be thought of as generalized theta functions (Beauville-Laszlo 93).

Specifically in the context of number theory/arithmetic geometry, by the theta function one usually means the Jacobi theta function (see there for more) for $z = 0$. While this is the historically first and archetypical function from which all modern generalizations derive their name, notice that at fixed $z$ as a function in $\tau$ the “theta function” is not actually a section of a line bundle anymore. The generalization in number theory of the Jacobi theta function that does again have a dependence on a twisting is the Dirichlet theta function depending on a Dirichlet character (which by Artin reciprocity corresponds to a Galois representation).

Certain integrals of theta functions yield zeta functions, see also at function field analogy.

context/function field analogytheta function $\theta$zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$)L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$)eta function $\eta$special values of L-functions
physics/2d CFTpartition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
Riemannian geometry (analysis)zeta function of an elliptic differential operatorzeta function of an elliptic differential operatoreta function of a self-adjoint operatorfunctional determinant, analytic torsion
complex analytic geometrysection $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$zeta function of a Riemann surfaceSelberg zeta functionDedekind eta function
arithmetic geometry for a function fieldGoss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number fieldHecke theta function, automorphic formDedekind zeta function (being the Artin L-function $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation)Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)class number $\cdot$ regulator
arithmetic geometry for $\mathbb{Q}$Jacobi theta function ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character)Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$)Artin L-function of a Galois representation $\mathbf{z}$ , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

### In quantization

Theta functions are naturally thought of as being the states in the geometric quantization of the given complex space, the given holomorphic line bundle being the prequantum line bundle and the condition of holomorphicity of the section being the polarization condition. See for instance (Tyurin 02). In this context they play a proming role specifically in the quantization of higher dimensional Chern-Simons theory and of self-dual higher gauge theory. See there for more.

Specifically the fact that in geometric quantization of Chern-Simons theory in the abelian case, and the holographically dual partition functions of the WZW model the choice of polarization is induced from the choice of complex structure $\mathbf{\tau}$ on a given Riemann surface $\Sigma$ and for each such choice there is then a section/partition function depending on a coordinte $\mathbf{z}$ in the Jacobian $J(\Sigma)$ is reflected in the double coordinate dependence of the theta function:

$\theta(\mathbf{z},\mathbf{\tau}) = \theta\left(gauge\;field\;configuration\;on\;\Sigma\;, \; complex\;structure\;on\;\Sigma\right) \,.$

Since from the point of view of Chern-Simons theory this is a wavefunction, one might rather want to write $\Psi(\mathbf{z},\mathbf{\tau})$.

For nonabelian CS/WZW theory the same story goes through and one may the elements of the corresponding conformal blocks “generalized theta functions” (Beauville-Laszlo 93).

## Definition

Consider a complex torus $T \simeq V/\Gamma$ for given finite group $\Gamma$.

Say that a system of multipliers is a system of invertible holomorphic functions

$e_\gamma \colon V \longrightarrow \mathbb{C}^\times \hookrightarrow \mathbb{C}$

satisfying the cocycle condition

$e_{\gamma + \delta}(z) = e_\gamma(z + \delta) e_\delta(z) \,.$

Then a theta function is a holomorphic function

$\theta \colon V \longrightarrow \mathbb{C}$

for which there is a system of multipliers $\{e_\gamma\}$ satisfying the functional equation which says that for each $z \in V$ and $\gamma \in \Gamma \hookrightarrow V$ we have

$\theta(z + \gamma) = e_\gamma(z) \theta(z) \,.$

e.g. (Beauville, above prop. 2.2), also (Beauville, section 3.4)

## Examples

The following table lists classes of examples of square roots of line bundles

## References

Introductions to the traditional notion include

• D.H. Bailey et al, The Miracle of Theta Functions (web)

• M. Bertola, Riemann surfaces and theta functions (pdf)

Modern textbook accounts include

Further discussion with an emphasis of the origin of theta functions in geometric quantization of Chern-Simons theory is in

• Arnaud Beauville, Theta functions, old and new, Open Problems and Surveys of Contemporary Mathematics SMM6, pp. 99–131 (pdf)

• Andrei Tyurin, Quantization, Classical and quantum field theory and theta functions, AMS 2003 (arXiv:math/0210466v1)

• Yuichi Nohara, Independence of polarization in geometric quantization (pdf)

• Gerard Lion, Michele Vergne, The Weil representation, Maslov index and theta series

Specifically the theta functions appearing in 2d CFT as conformal blocks and as spaces of sections of prequantum line bundles in quantization of Chern-Simons theory are discussed for instance in

and more generally the partition functions of connection-twisted Dirac operators on even-dimensional locally symmetric spaces is discussed in

Generalization of this from abelian to non-abelian conformal blocks to “generalized theta functions” appears in

brief review is in

• Krzysztof Gawedzki, section 5 of Conformal field theory: a case study in Y. Nutku, C. Saclioglu, T. Turgut (eds.) Frontier in Physics 102, Perseus Publishing (2000) (hep-th/9904145)

That the Riemann zeta functions are the local coordinate expressions of the covariantly constant sections of the Hitchin connection is due to

• Nigel Hitchin, remark 4.12 in Flat connections and geometric quantization, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. (Euclid)

Relation to elliptic genera (see also at Jacobi form)

• Kefeng Liu, section 2.4 of On modular invariance and rigidity theorems, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 (EUCLID, pdf)

Theta functions for higher dimensional varieties and their relation to automorphic forms is due to

• André Weil, Sur certaines groups d’operateur unitaires, Acta. Math. 111 (1964), 143-211

see Gelbhart 84, page 35 (211) for review.

Further developments here include

• Stephen Kudla, Relations between automorphic forms produced by theta-functions, in Modular Functions of One Variable VI, Lecture Notes in Math. 627, Springer, 1977, 277–285.

• Stephen Kudla, Theta functions and Hilbert modular forms,Nagoya Math. J. 69 (1978) 97-106

• Jeffrey Stopple, Theta and $L$-function splittings, Acta Arithmetica LXXII.2 (1995) (pdf)

• Yum-Tong Siu, Theta functions in higher dimensions (pdf)

Revised on July 19, 2015 13:11:54 by Urs Schreiber (94.118.157.94)