Would it surprise you to know that one in three women worldwide have experienced physical or sexual violence from their intimate partner? Or that as many as 38% of women who are murdered globally are killed by their partners? It is a sad reality, but those are the facts.
Globally, the most common form of violence against women is from an intimate partner. The statistics are shocking. And while these numbers are widely disseminated, the facts persist. The stories repeat themselves, affecting girls and women around the world regardless of race, nationality, social status or income level.
This sad reality was the cause of Nahr Ibrahim Valley's death in Lebanon, just months after the country's new law on domestic violence was finally passed. The new law came after several cases sparked campaigns and protests in the Lebanese capital surrounding International Women's Day last year. Unfortunately, it was not enough to save her life, but it can be the hope for thousands of women in the country, who previously had no legal protection against this type of crime.
The World Bank Group's Women, Business and the Law project studies where countries have enacted laws protecting women from domestic violence. The fourth report in the series, Women, Business and the Law 2016: Getting to Equal, finds that more than 1 out of 4 countries covered around the world have not yet adopted such legislation. The effects of this form of violence are multifold. It can lead to lower productivity, increase absenteeism and drive up health-care costs. Moreover, where laws do not protect women from domestic violence, women are likely to have shorter life spans.
Domestic violence, also viewed as gender-specific violence, commonly directed against women, which occurs in the family and in interpersonal relationships, can take different forms. Abuse can be physical, emotional, sexual or economic. The 2016 edition of Women, Business and the Law shows that, even where laws do exist, in only 3 out of 5 economies do they cover all four of those types of violence. Subjecting women to economic violence, which can keep them financially dependent, is only addressed in about half of the economies covered worldwide.
As with most present-day facts, this issue has historical roots, largely based on social and cultural norms. Before the 1900s, in most cultures, husbands had the right – by law – to use violence to enforce authority over their wives. The husband's right to "physically discipline his wife" started slowly being removed in the late 19th Century. In 1878, the Matrimonial Causes Act in the United Kingdom made it legal for women victims of violence in marriage to obtain separation orders and, by the end of the 1870s, most United States courts had rejected the right of husbands to chastise their wives. In the 1970s, specific laws against domestic violence started being introduced, but it was only in the 1990s that they started gaining strength worldwide, largely driven by international and regional human-rights conventions and campaigns.
Around the world, increased awareness and a rising outcry surrounding cases of domestic violence call for the enactment of more and better laws. Legal protection against domestic violence in its various forms is crucial to reduce impunity and to open avenues for redress.
On the extreme end of this spectrum are countries that still legally allow for domestic violence. As recently as 2010, a court in the United Arab Emirates ruled that a husband is permitted to beat his wife as long as he leaves no marks. But awareness has been growing around the issue, and campaigns pushing for the enactment of laws and for the elimination of violence against women are increasingly prominent.
October is National Domestic Violence Awareness month in the United States. While the passage of the Violence Against Women Act (VAWA) has enabled women to seek support and judicial response, cases of domestic violence are still widespread. The economic costs are also huge: Each year in the United States, eight million days of paid work are lost to domestic violence, amounting to some $8.3 billion in expenses. The adoption of the Violence Against Women Act (VAWA), as with similar laws around the world, can change people's views regarding the seriousness of the offense and can enable victims to obtain support.
The good news is that progress is continuing. In addition to Lebanon, within the last two years Belarus, Latvia, Saudi Arabia and Tonga enacted laws on domestic violence for the first time. And other countries – such as Italy and Macedonia – have amended their legislation to cover economic violence.
Campaigns and awareness-building around the world have played a critical role in promoting the enactment of laws and behavior change over the past 25 years. Let's continue the work to enable even better results toward eradicating all forms of violence against women in the next 25.


text:
WEG supplies product package for a solar and wind power plant in Brazil
Five solar and wind power projects will be benefited by this product supply.
WEG contributes to the increase of renewable energy sources in the world, providing solutions for the segment, as well as expanding its market share at global level. An example of this was a supply of step up substations and transformer made to a multinational energy sector company for five solar and wind power plants of more than 962.7 MW.
The substations provided by WEG for these five projects are complete turnkey solutions including preliminary study about the project, definition of a customer-dedicated solution, project management, construction and assembly. For customer satisfaction, WEG counts of a qualified and trained staff with proven experience, having delivered and operated more than 430 substations at all voltage levels up to 550 kV.
In addition to complete and customized delivery, turnkey substations offer significant advantages such as interface with other products that WEG manufactures, such as HV switches, HV transformers, MV/LV oil and dry transformers, panels - control/protection and auxiliary services, cabinets - control/protection of feeder bays, monitoring system and generators.
According to ONS (Brazil´s Electric Grid Operator), every transmission and power generation substation interconnected to SIN (National Interconnected System) system must include two energy sources for essential loads. So, in addition to substations and transformers, WEG has also supplied 69 kVA and 200 kVA, IP44 Degree of Protection synchronous alternators for these projects. The generator set will operate at an emergency/stand-by service duty to cover unexpected failures of power outages from the electricity grid, keeping in operation essential equipment such as computers, control monitoring systems, control panels, among others.
What allows WEG to be successful with this type of supply is its verticalized production process, combined with the customization of projects, which provide customers with high flexibility and delivery speed up. Associated to this, state-of-the-art equipment a qualified technical team and a comprehensive product portfolio ensure added value to each solution designed for the customer.
WEG supplies substations for solar power plant in Mexico
The energy produced will be connected to the electricity grid of the Federal Electricity Commission (CFE), Mexico's state-owned electric utility.
WEG alternators are supplied to the Brazilian power utility CEMIG
This package of equipment is part of generator sets for several CEMIG's projects
WEG and Engie Brasil Energia complete installation of the first national wind turbine
Wind turbine is part of the project for the first National Wind Turbine, which is intended to develop national technology and production chain for the manufacture and installation of large size equipment in the country

redpajama_set_name: RedPajamaCommonCrawl


what about the dark version for Windows 10 ?
Love this skin. However I noticed the windows explorer Minimize, Restore, and Close buttons at the top of every window are not responding to mouse correctly. When I mouse over Minimize, the Restore is actually highlighted until I mouse the mouse even further left toward the center of Minimize. Same with Restore – Close is highlighted. So all response seems off center, and often times I will accidentally close a window rather than Restore it, or Restore it rather than Minimize it. If there is a fix, let me know how, because I love the gray tones and icons and everything else about this skin.
1. How can delete the gnome "orb" only or restore the original orb windows ?
2. I can't install the gnome skin pack. always i try to uninstall but after restart, "windows startup repair" is launching, causing restore the skinpack again before the uninstall.
hi during uninstalation i didnt tick the file option. Skin is completely removed but star menu button, folders apperarnce and other things remain as it is…please help me how can i restore default of all these??
How I can only change icons for paritition, dvd drive, removable disks…? And were I can download icons like this in this theme?
why its so hard to download..I'm frust waiting..
thnx, i will try to uninstall.
my window explorer kept crash n need restart every time i restart my pc. and i cant even open any type of game because of this prob too.. can i know where is not right?
1. rocketdock settings could not be saved… I have to manually start with administrator privileges every time..
2. boot screen is not showing properly(where 'starting Gnome' appears).. only cyclic animation is showing..


Just down from the slopes of the Alps? You look amazing. Now, next up on the itinerary, pop-in to any of these four Italian capitals-of-style to enjoy a cozy luxurious stay. And before you go, perhaps check our list of iconic films that are so associated with each locale.
What to See: Ancient Rome and its incredible iconic structures, especially if there is snow.
City Film: We can't but help imagine Milan as portrayed in the opening wintery scenes of the film, I Am Love (2010).
The newly restored Galleria Vittorio Emanuelle II. This glass-roofed iconic structure is filled with elegant shops and a few cafes. Right next door is the Duomo.
City Film: Casino Royale (2006) we love it, and it shows off the city to spectacular effect, but on a more intimate level try Brideshead Revisited, also (2008).
Carnevale should be in full swing (January 23rd - February 1st). This is also the season of the Acqua Alta (high water), so be prepared to walk on the city's other type of famous bridges, those elevated wooded paths which will to keep your feet dry.
City Film: The iconic A Room with a View (1985) and the terrifying Hannibal (2001).
It's cold in Florence in the winter (Wow, thanks, Red Maps, big surprise!) so take the opportunity without the summer crowds to see the Uffici and Accademia Museums.
Images: top d'Inghilterra Rome; Danieli Venice; below left Armani Milan; below right Lungarno Suites Florence; bottom Bulgari Milan.


Any time you purchase goods that must be delivered to you through the mail or a delivery service (such as when you buy an item online), there could be a shipping & handling charge. This charge is meant to cover the packing and postage or shipping costs for getting the item to you.
There are a lot of different ways that merchants charge for shipping and handling. Some build the costs right into the price of the goods. Others charge a flat rate (one constant fee) for all purchases. Some use a formula that you can use to help predict the final cost of your purchase. This lesson will focus on this last type, where we use a formula.
When using a formula, there is frequently a minimum charge that is the least you will pay for shipping and handling on small orders. If the calculated shipping and handling charge of the order is below the minimum, then the shipping and handling charge will equal the minimum charge.
If the order is more than the minimum, a shipping and handling charge could be calculated as a percent of the total purchase price or the weight of the goods (weight is often used by delivery services and the U.S. Postal Service to determine their prices, but formulas on prices might be easier to manage by the merchant).
For example: Company ABC has a $2.00 minimum charge for shipping and handling, and a rate of 10% of the total order price.
If you make an order of $15, then the calculated charge is $15*0.10 = $1.50, but that is below the minimum of $2, so the shipping and handling charge will be $2.
If you make an order of $30, then the shipping and handling charge will be $30*0.10 = $3.
What is the shipping and handling charge for a $ order with a rate of % and a minimum charge of $.00?


French-Chinese model, Estelle Chen won France's Elite Look Model competition in 2013 and has taken her time to build a strong portfolio while finishing school. She made her first international appearance at the Paris Fashion Week last year and the following season, she showed up on the runways at Louis Vuitton, Miu Miu, Kenzo and Loewe. This Vogue Italia Beauty editorial is her first in a major fashion publication. Congratulations, Estelle!


Other Attendees: Andy Jansky, Jeremy O'Leary, Mary Vogel, Mary Ann Schwab, Jerry Grossnickle- Forest Park NA, other members of Forest Park NA, Corbett-Terwilliger-Lair Hill NA member, Michele Crim – BPS, Nan Stark – BPS, Marty Stockton – BPS, Joan Frederiksen – BPS, Bob Glascock – BPS, Sandra Wood – BPS.
View the original agenda, including materials, for this meeting.
BPS Staff Radcliffe Dacanay gave a brief presentation on Portland demographics to enhance PEG members' understanding of equity considerations in land use planning. PBEM Director Carmen Merlo and BPS Staff Roberta Jortner gave a presentation on Natural Hazards policy issues and preliminary draft goals and policies. PEG members also received a presentation from BPS staff Tom Armstrong, lead of the Economic Development PEG, on policy regarding development of industrial lands.
Overall, PEG members urged Staff to strengthen draft policies regarding reducing risks and impacts of natural hazards, and staff to strengthen policy language to ensure commitments to action.
Summary: Dena welcomes the group, requests any photos or bios that have not been submitted, and then asks the PEG to consider a 3:30 to 6:30 time frame for the September 27th meeting.
Summary: When considering Comp Plan policy recommendations through an equity lens, PEG's and City planners are encouraged to study the regional distribution of racial demographics, income, education levels, and poverty distributions throughout the city.
Education, income levels, and racial demographics are closely tied together, and sometimes correspond with natural hazard zones.
A regional hub,Portlandcentral city will see increased plans to develop infrastructure and public transportation.
Summary: An important objective of The Comprehensive Plan is to incorporate goals and policies that will reduce the risks and impacts of natural hazards on people, property and ecological systems. The current Comprehensive Plan does not contain a cohesive set of policies addressing natural hazards. While existing City codes (flood, seismic, fire) do address natural hazards considerable risks continue to exist.
Staff drafted new policies to protect natural resource systems, reduce structural and non-structural hazards, address infrastructure constraints, and promote emergency preparedness. The intent is that these policies will help inform future plans, codes and investments to make the City more resilient in the event of a natural disaster. In addition, the City needs to consider balancing livability factors (e.g., trees in neighborhoods) with promoting effective access to emergency services.
General consensus to use stronger language in policies: e.g. replace "encourage" with stronger words; use words such as reduce, avoid, commit.
Regarding specific policies, see matrix and detailed discussion in Flip Chart notes at end of this summary.
Are there things that we would want to embody in this plan that would help us though something catastrophic? Take a more proactive response on some of these policies. Do we need a policy on how we incorporate new information on hazards?
Need to provide for alternatives in emergency and hazard planning. Important that we recognize varying levels of risk and it's not a one size fits all approach to these hazards.
Need to try to align investments that don't create new problems, but are designed to solve problems with compatible strategy; multiple problems overlap and our strategies should address as many problems as possible.
Do we have an inventory of what has been mitigated due to development, and cost? This may inform a lot of what was said.
Creating partnerships when we talk about flood control. How do we utilize that and fit it into the inventory of lands available? How do we handle multiple hazard zones?
Summary: Portland is the regional job center with 39% of the regional jobs and 26% of the population. BPS is attempting to plan for 147,000 new jobs to the city over the next 25 years. The EcDev PEG has been discussing the issues of shortfalls of available land in industrial and institutional areas. The WHE PEG is encouraged to consider what can be done to encourage development on existing industrial parcels? What can we do to expand that industrial land capacity?
Concern that by the time the Watershed PEG has their next meeting the EcDev PEG will have already weighed in on their recommendations.
If you have incentives to develop green fields then how are we ever going to recover the brownfields?
An e-mail will be sent out to schedule a 3:30 - 6:30pm meeting on September 27th.
Please help us finish the PEG bios by submitting your statements and photos.
For more information, please contact Sallie Edmunds, Bureau of Planning and Sustainability at 503-823-6950 or sallie.edmunds@portlandoregon.gov or Dena Marshall, Facilitator at 503-249-0000 or dena@decision2.com.
Are the proposed goals and policies provided appropriate and sufficient enough to address natural hazards in the Comp Plan? Doe they contribute positively or negatively to environmental justice and social equity?
What are the implications of directing future development away from hazard-prone areas?
New development vs. redevelopment and alterations?
Separate set of goals for reconstruction after a major disaster?
Balance w/"not negatively impacting job growth in disadvantaged communities", clarify which natural hazards policy is mitigating for; 404 Floodplain mitigation, other?
Design w/nature "supporting development capacity" to not be adversely affected? Or convert to natural stormwater systems? - restore to hydrology. See attachment B – hydrology goal – stormwater management.
Add native vegetation, corridors, permeability, connectivity.
Needs to be a holistic approach. How is it going to affect the larger area?
Common terms needed among policy makers at state and local levels.
What does encourage mean; how will landscape look with climate change.
Mitigation banks have been a mixed review. Now, encourage innovation and resiliency in policy.
Wetlands – if unmanaged, become a hazard; need a management plan beyond 5 years, holistic; responsibility to maintain.
First step should be to avoid, then find innovative approaches for mitigation.
Concern about "innovative mitigation" - need to be mindful of concentrating natural resources on the landscape. Need a policy to address distribution, dispersal?
Policies too weak: Change "encourage" a assure or support.
Flood storage is a natural feature - protection needs to be addressed more explicitly. Identify need for natural buffers.
Built environment - examine and be proactive with viable uses.
Policies for a do-over in the event of a traumatic event or natural disaster? Do we need a policy for incorporating new information on hazards?
Priorities for policies do-over, e.g. development in the floodplain.
More clearly define what's being mitigated and what for.
Acknowledge that some development needs to occur in hazard areas (thoroughfares, airports in floodplains) à focus land use planning on preparedness.
Different approaches / strategies for different hazards: Retrofitting, evacuation, etc.
Call for specific language to align concepts and strategies so that they are complementary (not contradictory), where the whole is greater than the sum of the parts".
Creating partnerships around flood control projects - how do private partnership mitigation efforts factor into it?
How to handle multiple hazard zones?
Priority recommendations for PBEM: (1) land use planning, zoning codes, building laws – to minimize impacts of natural hazards.
Centennial neighborhood and others - curb density especially where public transportation does not meet the need.
Forest Park NA –creating policy amendment to Comp Plan for wildlife deferral (not forest deferral). The wildlife corridor connects the coast line to Forest Park; we've been very proactive in trying to prevent development around the park.
Treating "p" zones and "c" zones the same? - No, 'p' zones dropped from buildable lands.
What is the end game? How do we get those extra seven hundred acres we need to make the city function?


Blanket Bay Camping, Great Ocean Road Great Otway National Park Victoria Australia, a popular campground where advance bookings are required year round and camping fees apply. Bookings open at 12n on the 1st of August each year for the peak summer holiday period 26 December to 26 January – maximum stay is 7 nights and a ballot system applies. Bookings open at 12 noon on 2 February for Labour Day weekend (minimum booking of 3 nights required), and for Easter (minimum booking 4 nights required). There are 22 campsites available at Blanket Bay Campground, each can accommodate a maximum of 6 people per site. All campsites are unpowered, and campsites are suitable for tents, camper-trailers, campervans, motorhomes and other recreational vehicles. There are non-flush toilets, a shared picnic table, beach access and shared fireplaces – fires strictly only in fireplaces provided, and you must supply your own firewood. Fuel stoves, such as a gas stove, is recommended for cooking. Some water is available but is not suitable for drinking; you must carry your own fresh drinking water. This is a very popular camping area with many visitors making bookings to come to Blanket Bay at the same time each Christmas and Easter. There are some good sites with shade near an ocean beach to the south-west of Apollo Bay. When camping at Blanket Bay, make sure you visit the historic Cape Otway Lightstation. Approximately 20 km west of Apollo Bay, turn onto Cape Otway Lighthouse Rd and then after 9 km, take the signposted turning onto Blanket Bay Rd and drive for another 6 km.


text:
Creating One Community
The guiding principle of 'One Community' is paramount in Toyota Australia's new workplace at Sydney Olympic Park. This new environment creates a community from two separate subsidiaries and Davenport Campbell worked to ensure that both teams felt at home in their new environment.
Working collaboratively with both teams, it became clear that a space that encourages a culture of social inclusion would be key to the success of the project. As a result, the communal and informal spaces function as the social glue between the two businesses and set an inclusive tone that is maintained throughout the workplace. In contrast, the team-based areas have been catered to suit individual business operations to offer function and privacy to both parties.
Toyota's brand guidelines are thoughtfully integrated alongside soft curves and a neutral palette. Raw textural elements are used to reference the local context, and the kitchen features a relaxed residential style that welcomes teams to make themselves at home.
Stay tuned for Stage 2 to see the complete project!


text:
Booking here
Top Things To Do And See In Ibiza
Cala Pada Beach is just one of the stunning white beach locations on the Spanish isle of Ibiza. One of the Balearic Islands, Ibiza is a popular holiday destination for families, couples and groups of friends, with many different vacation styles to offer. Here is a guide for what to know about Ibiza before you go.
Visit the capital city
Ibiza Town is home to Ibiza Cathedral, known as Catedral de la Verge de les Neus and is dedicated to Our Lady of the Snows. It is in Dalt Vila, the ancient walled fortress of the city and it is one of the most popular things to do in Ibiza Town. A UNESCO World Heritage site, the Old Town is best explored on foot as it is made up of winding, tiny cobbled streets. If you are looking for things to do in Ibiza, then head here. It's warm enough to see the old Arabic roots of the city, the castle of Ibiza and for those who love art, the Museu Puget which has a permanent collection of artwork by Ibizan born artists Narcis Puget Vinas (1874 – 1960) and his son Narcis Puget Riquer (1916 – 1983). The views from this part of town are stunning and include a view out to Ibiza airport, Playa d'en Bossa, Marina Botafoch and even to the smaller Balearic Island of Formentera.
Hit the dancefloor
No guide to Ibiza is complete without mentioning it's world-famous nightlife. Nightlife in Ibiza is not just Pacha and Amnesia nightclubs although these are packed out every night during the clubbing season which runs from May to the end of September. Classic nights include Cream at Amnesia and celebrity spotting at Pacha. One of the top things to do in Ibiza in September is to go to a closing party at one of these top clubs. For those who are looking for something more chilled and things to do with family and friends, or to enjoy an evening meal or just drinks, the best places to go are Playa d'en Bossa, San Rafael and Es Caná, which is also the best place to buy souvenirs at the Es Caná Hippy Market.
Get back to nature
Take a dip in the sea or go Scuba diving off the coast. Water activities are easy to come across on Ibiza with registered companies offering the safest experiences. Two famous sites for diving are the Don Pedro shipwreck, only five minutes dive away from Ibiza Town. The Cave of Light, located between Ibiza Town and Portinatx is the perfect dive spot for those who like to weave in and out of caves as they scuba dive. Dive here on a sunny day to spot a ray of sunlight through the rocks. Go walking Ibiza on the seashore for the best natural views.
Where to stay on the White Isle
Located just 500 meters from Santa Eulalia with an enviable location next to the sea front, the Catalonia Ses Estaques is an Ibiza all-inclusive resorts adult only hotel. It has a swimming pool, and for those who like to relax in a jacuzzi there are two outside wood-decorated Jacuzzis and sunbeds by the shoreline for topping up that tan.
Seafood is an excellent choice in the hotel's à la carte restaurant and is best enjoyed with a glass of chilled white wine.
The Catalonia Royal Ses Savines is located in a pine forest with stunning views of the Mediterranean Sea. It has two swimming pools, one of which includes a chill out area with sun loungers, Balinese beds and a bar which serves soft drinks and water.
ibiza ibiza beach santa eulalia ibiza
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text:
Q: Determinant connection between Schur polynomials and power sum polynomials Let $f_i=f_i(x_1,x_2,\ldots, x_n),i=0,1,2, \ldots $ be a family of symmetric polynomials. For the partition $\lambda=(\lambda_1,\lambda_2, \ldots, \lambda_n)$   consider the determinant
$$
D_\lambda(f)=\left | \begin{array}{lllll}
f_{\lambda_1} & f_{\lambda_1+1} & f_{\lambda_1+2} & \ldots & f_{\lambda_1+n-1}\\
f_{\lambda_2-1} & f_{\lambda_2} & f_{\lambda_2+1} & \ldots & f_{\lambda_2+n-2}\\
\vdots & \vdots  & \vdots  & \ldots & \vdots \\
f_{\lambda_n-(n-1)} & f_{\lambda_n-(n-2)} & f_{\lambda_n-(n-3)} & \ldots & f_{\lambda_n}
\end{array} \right|.
$$
It is well known (Jacobi−Trudi formulas) that for the elementary symmetric polynomials  $e_i=e_i(x_1,x_2,\ldots, x_n)$ and for the complete homogeneous symmetric polynomials  $h_i=h_i(x_1,x_2,\ldots, x_n)$  we have
$$
D_\lambda(h)=s_{\lambda}(x_1,x_2,\ldots, x_n) \text{ and } D_\lambda(e)=s_{\lambda'}(x_1,x_2,\ldots, x_n),
$$
where $s_{\lambda}(x_1,x_2,\ldots, x_n)$ is the Schur polynomial and $\lambda'$ is the conjugate partition.
Question. Is there a similar expression for $D_\lambda(p)$ where $p_i=p_i(x_1,x_2,\ldots, x_n)$ is the power sum symmetric polynomials?
By direct calculation for $n=2, \lambda_2\geq 1$ I got that
$$D_{(\lambda_1,\lambda_2)}(p)=-s_{(\lambda_1-1,\lambda_2-1)}  V(x_1,x_2)^2$$ and for $n=3,\lambda_3\geq 2$
$$D_{(\lambda_1,\lambda_2,\lambda_3)}(p)=-\frac{s_{(\lambda_1-1,\lambda_2-1,\lambda_3-1)}}{s_{(1,1,1)}}  V(x_1,x_2,x_3)^2$$
and so on.
Here $V$ is the Vandermonde determinant.
I hope that must be a nice   formula and for any $n$.

A: I don't know a fully general result, but your pattern for partitions $\lambda$
of length $\leq n$ with $n$-th entry $\lambda_{n}\geq n-1$ and with $n$
indeterminates persists:

Theorem 1. Let $n$ be a positive integer. Let $\lambda=\left(  \lambda
_{1},\lambda_{2},\ldots,\lambda_{n}\right)  $ be an integer partition with at
most $n$ parts. Assume that $\lambda_{n}\geq n-1$. Consider polynomials in $n$
indeterminates $x_{1},x_{2},\ldots,x_{n}$. For each nonnegative integer $k$,
we set
\begin{align*}
p_{k}:=x_{1}^{k}+x_{2}^{k}+\cdots+x_{n}^{k}.
\end{align*}
(This is the $k$-th power-sum symmetric polynomial in $x_{1},x_{2}
,\ldots,x_{n}$ when $k>0$. We have $p_{0}=n$.) Define the $n\times n$-matrix
\begin{align*}
P:=\left(  p_{\lambda_{i}-i+j}\right)  _{1\leq i\leq n,\ 1\leq j\leq
n}=\left(
\begin{array}
[c]{cccc}
p_{\lambda_{1}} & p_{\lambda_{1}+1} & \cdots & p_{\lambda_{1}+n-1}\\
p_{\lambda_{2}-1} & p_{\lambda_{2}} & \cdots & p_{\lambda_{2}+n-2}\\
\vdots & \vdots & \ddots & \vdots\\
p_{\lambda_{n}-n+1} & p_{\lambda_{n}-n+2} & \cdots & p_{\lambda_{n}}
\end{array}
\right)  .
\end{align*}
Let $\mu=\left(  \mu_{1},\mu_{2},\ldots,\mu_{n}\right)  $ be the partition
defined by
\begin{align*}
\mu_{i}=\lambda_{i}-\left(  n-1\right)  \ \ \ \ \ \ \ \ \ \ \text{for each
}i\in\left\{  1,2,\ldots,n\right\}  .
\end{align*}
(This is indeed a partition, since $\mu_{n}=\underbrace{\lambda_{n}}_{\geq
n-1}-\left(  n-1\right)  \geq0$.) Let $s_{\mu}$ be the corresponding Schur
polynomial in the $n$ indeterminates $x_{1},x_{2},\ldots,x_{n}$. Furthermore,
let
\begin{align*}
V_{n}:=\prod_{1\leq i<j\leq n}\left(  x_{i}-x_{j}\right)
\end{align*}
be the Vandermonde determinant. Then,
\begin{align*}
\det P=\left(  -1\right)  ^{n\left(  n-1\right)  /2}s_{\mu}\cdot V_{n}^{2}.
\end{align*}

Proof. Let $A_{\mu}$ be the $n\times n$-matrix
\begin{align*}
\left(  x_{j}^{\mu_{i}+n-i}\right)  _{1\leq i\leq n,\ 1\leq j\leq n}=\left(
\begin{array}
[c]{cccc}
x_{1}^{\mu_{1}+n-1} & x_{2}^{\mu_{1}+n-1} & \cdots & x_{n}^{\mu_{1}+n-1}\\
x_{1}^{\mu_{2}+n-2} & x_{2}^{\mu_{2}+n-2} & \cdots & x_{n}^{\mu_{2}+n-2}\\
\vdots & \vdots & \ddots & \vdots\\
x_{1}^{\mu_{n}+n-n} & x_{2}^{\mu_{n}+n-n} & \cdots & x_{n}^{\mu_{n}+n-n}
\end{array}
\right)  .
\end{align*}
It is then well-known that
\begin{equation}
s_{\mu}=\dfrac{\det\left(  A_{\mu}\right)  }{V_{n}}
.
\label{darij1.eq.slam=frac}
\tag{1}
\end{equation}
Indeed, this is the alternant formula for Schur polynomials. For a proof, see,
e.g., Corollary 2.6.7 in the lecture notes Darij Grinberg and Victor Reiner,
Hopf Algebras in Combinatorics,
arXiv:1409.8356v7. (The notations in those
notes are not quite ours. Namely, our matrix $A_{\mu}$ is the transpose of the
matrix whose determinant is $a_{\mu+\rho}$ in the notes, whereas our $V_{n}$
is $a_{\rho}$ in these notes. Corollary 2.6.7 has to be applied to $\mu$
instead of $\lambda$.)
Let $B$ be the $n\times n$-matrix
\begin{align*}
\left(  x_{i}^{j-1}\right)  _{1\leq i\leq n,\ 1\leq j\leq n}=\left(
\begin{array}
[c]{cccc}
1 & x_{1} & \cdots & x_{1}^{n-1}\\
1 & x_{2} & \cdots & x_{2}^{n-1}\\
\vdots & \vdots & \ddots & \vdots\\
1 & x_{n} & \cdots & x_{n}^{n-1}
\end{array}
\right)  .
\end{align*}
The Vandermonde determinant formula yields
\begin{align*}
\det B  & =\prod_{1\leq i<j\leq n}\underbrace{\left(  x_{j}-x_{i}\right)
}_{=-\left(  x_{i}-x_{j}\right)  }=\prod_{1\leq i<j\leq n}\left(  -\left(
x_{i}-x_{j}\right)  \right)  \\
& =\left(  -1\right)  ^{n\left(  n-1\right)  /2}\underbrace{\prod_{1\leq
i<j\leq n}\left(  x_{i}-x_{j}\right)  }_{=V_{n}}=\left(  -1\right)  ^{n\left(
n-1\right)  /2}V_{n}.
\end{align*}
However, we have
\begin{equation}
A_{\mu}B=P.
\label{darij1.eq.AB=P}
\tag{2}
\end{equation}
(Indeed, for any $i,j\in\left\{  1,2,\ldots,n\right\}  $, the $\left(
i,j\right)  $-th entry of the matrix $A_{\mu}B$ is
\begin{align*}
\sum_{k=1}^{n}\underbrace{x_{k}^{\mu_{i}+n-i}x_{k}^{j-1}}_{\substack{=x_{k}
^{\mu_{i}+n-i+j-1}=x_{k}^{\lambda_{i}-i+j}\\\text{(since }\mu_{i}=\lambda
_{i}-\left(  n-1\right)  \text{ and}\\\text{thus }\mu_{i}+n-i+j-1=\lambda
_{i}-\left(  n-1\right)  +n-i+j-1=\lambda_{i}-i+j\text{)}}}  & =\sum_{k=1}
^{n}x_{k}^{\lambda_{i}-i+j}\\
& =x_{1}^{\lambda_{i}-i+j}+x_{2}^{\lambda_{i}-i+j}+\cdots+x_{n}^{\lambda
_{i}-i+j}=p_{\lambda_{i}-i+j},
\end{align*}
which happens to be precisely the $\left(  i,j\right)  $-th entry of the
matrix $P$. Thus, \eqref{darij1.eq.AB=P} follows.)
Now, the two matrices $A_{\mu}$ and $B$ are square matrices. Hence,
\begin{align*}
\det\left(  A_{\mu}B\right)    & =\underbrace{\det\left(  A_{\mu}\right)
}_{\substack{=s_{\mu}V_{n}\\\text{(by \eqref{darij1.eq.slam=frac})}}
}\cdot\underbrace{\det B}_{=\left(  -1\right)  ^{n\left(  n-1\right)  /2}
V_{n}}\\
& =s_{\mu}V_{n}\cdot\left(  -1\right)  ^{n\left(  n-1\right)  /2}V_{n}=\left(
-1\right)  ^{n\left(  n-1\right)  /2}s_{\mu}\cdot V_{n}^{2}.
\end{align*}
In view of \eqref{darij1.eq.AB=P}, we can rewrite this as
\begin{align*}
\det P=\left(  -1\right)  ^{n\left(  n-1\right)  /2}s_{\mu}\cdot V_{n}^{2}.
\end{align*}
This proves Theorem 1. $\blacksquare$
The claim of Theorem 1 can further be rewritten by observing that (in $n$
indeterminates $x_{1},x_{2},\ldots,x_{n}$) we have
\begin{align*}
s_{\lambda}=s_{\mu}\cdot\left(  x_{1}x_{2}\cdots x_{n}\right)  ^{n-1}
\end{align*}
(because the entries of $\lambda$ are the respective entries of $\mu$ plus
$n-1$). The product $x_{1}x_{2}\cdots x_{n}$ can also be rewritten as
$s_{\left(  1^{n}\right)  }$, where $\left(  1^{n}\right)  $ is the partition
$\left(  1,1,\ldots,1\right)  $ with $n$ entries.

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text:
Syntaktyka, składnia (gr.  sýntaxis – "porządek, szyk",  syntaktikós – "porządkujący") – dział językoznawstwa, który zajmuje się budową wypowiedzeń.

Syntaktyka (w ujęciu tradycyjnym) bada takie zjawiska jak funkcje wyrazów w zdaniu (funkcja podmiotu, orzeczenia, dopełnienia itd.), zależności między wyrazami w zdaniu (np. związek zgody, związek rządu, związek przynależności) czy szyk wyrazów w wypowiedzeniu.

W odmiennym ujęciu (występującym w gramatyce generatywnej) syntaktyka opisuje sposoby łączenia się morfemów w zdania. W ujęciu gramatyki generatywnej syntaktyka przeciwstawia się semantyce i fonologii, morfologia natomiast stanowi część fonologii i syntaktyki.

Zobacz też

 syntaktyka w logice, zwana też czasem składnią logiczną
 składnia dialektalna (gwarowa)

Przypisy

!
Działy językoznawstwa

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