Meromorphic Functions Sharing the Same Zeros and Poles

Meromorphic Functions Sharing the Same Zeros and Poles

Hinkkanen’s problem (1984) is completely solved, i.e., it is shown that any meromorphic function f of
one complex variable is determined by its zeros and poles and the zeros of f(j), for j = 1, 2, 3, 4.

Theorem 1.1 (Nevanlinna’s Five-value Theorem (Nevanlinna 1929)) If two meromorphic functions share
five values IM, then they are equal.
The number five of IM shared values cannot be reduced.
Example 1 Let f(z) = exp(z) and g(z) = exp(−z). Then f and g share four values 0, 1, -1, ∞ IM. But f and g are
Theorem 1.2 (Nevanlinna’s Four-value Theorem (Nevanlinna 1929)) If two meromorphic functions, f and
g, share four values CM, then f is a M¨obius transformation of g.

Tranquilpeak — A gorgeous responsive theme for Hugo blog framework

Tranquilpeak — A gorgeous responsive theme for Hugo blog framework


Hugo version of Tranquilpeak is a based on original Hexo version This version is simply a port to Hugo static site generator.

Please all the credit should be attributed to original Hexo version and its author Louis Barranqueiro.

Hugo version keeps every .js and .css files untouched from original Hexo version in order to enjoy futur original Hexo version updates or features!


Quick start

Authors: Louis Barranqueiro (LouisBarranqueiro) and Thibaud Leprêtre (kakawait)
Version: 0.3.1-BETA (based on Hexo version 1.9.1)
Compatibility: Hugo v0.20.1

General features:

Fully responsive
Optimized for tablets & mobiles
Configurable menu of the sidebar
Pages to filter tags, categories and archives
Background cover image
Beautiful about page
Support Open Graph protocol
Easily customizable (fonts, colors, layout elements, code coloration, etc..)
Support internationalization (i18)
Posts features:

Thumbnail image
Cover image
Responsive videos & images
Sharing options
Navigation menu
GitHub theme for code highlighting (customizable)
Image gallery
Tags for images (FancyBox), wide images, tabbed code blocks, highlighted text, alerts
Table of contents
Integrated services:

Google analytics
Facebook Insights

Academic — Create a beautifully simple personal or academic website

Academic — Create a beautifully simple personal or academic website

Academic is a framework to help you create a beautiful website quickly. Perfect for personal, student, or academic websites. Check out the latest demo of what you’ll get in less than 10 minutes.

Key features:

Easily manage your homepage, blog posts, publications, talks, and projects
Configurable widgets available for Biography, Publications, Projects, News/Blog, Talks, and Contact
Need a different section? Just use the Custom widget!
Write in Markdown for easy formatting and code highlighting, with LaTeX for mathematical expressions
Social/academic network linking, Google Analytics, and Disqus comments
Responsive and mobile friendly
Simple and refreshing one page design
Easy to customize The Decline of Marriage and Rise of New Families The Decline of Marriage and Rise of New Families

Over the past 50 years, a quiet revolution has taken place in this country. Decades of demographic, economic and social change have transformed the structure and composition of the American family. The pre-eminent family unit of the mid-20th century—mom, dad and the kids—no longer has the stage to itself. A variety of new arrangements have emerged, giving rise to a broader and evolving definition of what constitutes a family.

At the center of this transformation is the shrinking institution of marriage. In 1960, 72% of American adults were married. By 2008, that share had fallen to 52%.

Part of this decline is explained by the fact that the average age at which men and women first marry is now the highest ever recorded, having risen by roughly five years in the past half century.1 And part of the decline is attributable to the near tripling in the share of currently divorced or separated—to 14% in 2008 from 5% in 1960.2

Public attitudes toward marriage reflect these dramatic changes. When asked in the new survey if marriage is becoming obsolete, about four-in-ten Americans (39%) say that it is. In a survey of voters conducted by Time magazine in 1978, when the divorce rate in this country was near an all-time high, just 28% agreed that marriage was becoming obsolete.3

Changes in marital patterns have had a major impact on the lives of children in this country. Marriage is no longer considered a prerequisite for parenthood. Over the past 50 years, the share of children born to unmarried mothers has risen dramatically—increasing eightfold from 5% in 1960 to 41% in 2008.This trend has contributed to the decrease in the share of children under age 18 living with two married parents – to 64% in 2008 from 87% in 1960.4

There are distinctive socio-economic, generational and racial patterns in the trends away from marriage and toward single parenthood and other emerging family forms.

Marriage rates are now more strongly linked to education than they have been in the past, with college graduates (64%) much more likely to be married than those who have never attended college (48%).

The racial differences are even larger. Blacks (32%) are much less likely than whites (56%) to be married, and this gap has increased significantly over time. And black children (52%) are nearly three times as likely as white children (18%) and nearly twice as likely as Hispanic children (27%) to live with one parent.

As the country shifts away from marriage, a smaller proportion of adults are experiencing the economic gains that typically accrue from marriage. In 2008, the median household income of married adults was 41% greater than that of unmarried adults, even after controlling for differences in household size.5 In 1960, this gap was only 12%. The widening of the gap is explained partly by the increased share of wives in the workforce (61% in 2008 versus 32% in 1960) and partly by the increased differential in the educational attainment of the married and the unmarried.6

The net result is that a marriage gap and a socio-economic gap have been growing side by side for the past half century, and each may be feeding off the other. As will be shown in greater detail in Chapter 2, adults on the lower rungs of the socio-economic ladder (whether measured by income or education) are just as eager as other adults to marry. But they place a higher premium on economic security as a prerequisite for marriage than do those with higher levels of income and education. And this is a bar that they—and their pool of prospective spouses—may find increasingly difficult to meet, given the fact that, relative to other groups, they have experienced significant economic declines in recent decades.

The changes in marriage rates are driven in large part by the behavior and attitudes of young adults, who are both delaying marriage and entering into less-traditional family arrangements. In 1960, 68% of adults ages 20-29 were married. By 2008, only 26% were married. The fact that young adults are delaying marriage does not necessarily mean they will never marry. Only time will tell. Meantime, it’s still the case that the vast majority of adults in the U.S. eventually get married. Among those ages 45 and older roughly nine-in-ten have ever married.

That said, young adults do have much different attitudes toward the trends that are driving family change. Nearly half of those under age 30 (46%) say the growing variety of family arrangements is a good thing, compared with just 30% of those ages 30 and older. In addition, young adults are much more accepting than their older counterparts of a host of societal trends affecting families, from more people living together without getting married to more gay and lesbian couples raising children.

Public Reactions to Decades of Change

The public is aware of the changes in marriage and family that have taken place over the past 50 years—and accepts some more readily than others. There is no clear consensus on the overall merit of the rise of new family arrangements. When asked whether the growing variety in the types of family arrangements is a good thing, a bad thing or doesn’t make a difference, the public is evenly split. A third (34%) say it’s a good thing, 29% say it’s a bad thing and 32% say it doesn’t make a difference.

Where you stand on this issue depends to some degree on how you live. Adults who are living a more traditional family life—married with children—are among the most resistant to the growing variety of family arrangements: 38% say it is a bad thing. Those who are divorced or separated have more of a live and let live attitude: a 39% plurality say the changes don’t make a difference. Those who are living with a partner are largely supportive of the new arrangements: a 56% majority says the growing variety is a good thing.

Of all the changes in family structure, the one that draws the strongest negative reaction is single parenthood. An overwhelming majority (69%) say the trend toward more single women having children without a male partner to help raise them is a bad thing for society. And a majority (61%) still believe that a child needs both a mother and a father to grow up happily.

Other changes get a much better reception from the public. For example, more than six-in-ten (62%) now say that the best kind of marriage is one where the husband and wife both work and both take care of the household and children. In 1977, fewer than half (48%) endorsed this egalitarian template for spousal roles.

And the public is quite open to the idea that marriage need not be the only path to family formation. An overwhelming majority says a single parent and a child constitute a family (86%), nearly as many (80%) say an unmarried couple living together with a child is a family, and 63% say a gay couple raising a child is a family.

Advanced Web Machinery

Advanced Web Machinery

Hello there, we are Dávid Csákvári and Tamás Sallai!

Dodie’s image Sashee’s image We are tech-savvy developers mainly on the web-dev field. We both hold a Master’s Degree in Computer Sciences and have more than half decade of field experience. We always seek new processes and technology to make better software. This blog is about the industry insight and problem solving experiences we encounter on a daily basis.

Better TeX math typesetting in Hugo

Better TeX math typesetting in Hugo

There is a page in the Hugo documentation that describes how to use MathJax to embed nicely-typeset mathematics in one’s Hugo-generated site.

For my own site, I took this as a starting point and made a few improvements. Here’s how I do the math typesetting in this blog.


I author my posts in Markdown, where the bulk of the content is plain text. When I want to include specially-typeset mathematical expressions, I use a custom shortcode to convert TeX-style equation definition into a snippet of HTML that looks nice when rendered in the browser.

So when I author a post, I’ll write something like this:

Here’s sum inline math: {{}}.

Display mode math looks like
Hugo will process this, apply my shortcode, and generate HTML that ultimately renders like this:

Here’s sum inline math: \sum_{n=1}^{\infty} 2^{-n} = 1∑
​​ 2
​​ =1 .

Display mode math looks like
\int \frac{1}{x} dx = \ln |x|∫

​​ dx=ln∣x∣
Shortcode and supporting markup

Here’s the definition of the shortcode:

<!– Hugo TeX shortcode
inline eqn: {{}}
display mode eqn: {{}}

{{ if .IsNamedParams }}
$${{ .Get “display” }}$$
{{ else }}
\({{ .Get 0 }}\)
{{ end }}

{{ if .IsNamedParams }}

{{ else }}

{{ end }}

This takes care of inserting the required content and tags for each particular expression, but that alone isn’t enough. Some additional Javascript and CSS are required in the header and footer of the page, as well.

In my Hugo page template I conditionally include this extra markup, which will pull down the CSS and Javascript libraries that do the real typesetting work when someone loads the page:

{{ if .GetParam “hasMath”}}
{{ end }}

{{ if .GetParam “hasMath”}}
{{ end }}
I simply add hasMath: true to the front matter of any post that has equations, and everything is taken care of. If a post doesn’t contain any mathematics, no superfluous scripts or CSS are included.

As you can see above, I am using KaTeX, a MathJax competitor from Khan Academy, as my backend typesetting library. KaTeX renders noticeably faster than MathJax, but has a more limited feature set. Given the meager sophistication of the equations appearing in this blog, I don’t need the more exotic features anyway, so KaTeX a pretty clear win.

Handling noscript

I’ve also taken care to include a block for each equation, which will activate when Javascript is disabled.

In such a case, KaTeX or MathJax won’t work, so I fall back to using flat images. Images don’t render as crisply, and won’t resize or zoom quite as well as proper HTML + CSS, but they will do in a pinch.

There are a few free cloud services available for this, which dynamically generate math typesetting images. You just embed the equation you want in the URL query parameters, and the server will create the image on the fly.

Google Charts is one such service. I’m currently using CodeCogs, as I find that its images look better than Google’s, and they support both inline and display-mode image rendering.

With Javascript disabled, the earlier example will render like this:

Here’s sum inline math: \sum_{n=1}^{\infty} 2^{-n} = 1.

Display mode math looks like

\int \frac{1}{x} dx = \ln |x|
To prevent the Javascript-only equation markup from appearing “raw” next to the images, I include a small bit of CSS in every page to hide such elements:

.jsonly { display: none }

RSS readers

Some people consume my blog through an RSS reader, rather than visiting my site directly. It would be nice if equations came through in a reasonable way for for these people, too.

Javascript is not exposed to RSS at all, so KaTeX and MathJax are out the window, and I am limited to HTML and inline CSS for formatting. Thankfully, this is exactly how the noscript fallback operates, so most of the work is already done.

After running Hugo to generate all of my site’s HTML and RSS content, I run a post-processing script to fix up the RSS XML slightly. The script does 2 things:

Removes the jsonly spans entirely, since they aren’t useful for RSS and will be rendered in their ugly raw form.
Removes the noscript opening and closing tags, leaving just the inner image content. Some RSS readers will not render content inside of noscript tags, so it’s important to remove these.
Here’s a snippet of PowerShell that does the trick:

$content = [io.file]::ReadAllText(‘index.xml’)

# delete jsonly spans completely
$content = $content -replace ‘<span class="jsonly">\s*.+?\s*</span>’,”

# remove tags, leaving just the inner content
$content = $content -replace ‘<noscript>\s*((?:.|\s)+?)\s*</noscript>’, ‘$1’

[io.file]::WriteAllText(‘index.xml’, $content)
What’s left should look fairly decent in most RSS readers.