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PASCAL’S MATRIX IN Zn

Speaker: TUYETDONG PHAN-YAMADA Department of Mathematics


PASCAL’S TRIANGLE

The formula for each element in row n is: đ?’? đ?‘›! đ?’?đ??śđ?’“ = = , where r = 0, 1, 2, ‌, n. đ?‘&#x;! đ?‘›âˆ’đ?‘&#x; ! đ?’“


THE SUMS OF THE ROW PATTERNS

Row 0: 20 = 1 Row 1: 21 = 1 + 1 = 2 Row 2: 22 = 1 + 2 + 1 = 4 Row 3: 23 = 1 + 3 + 3 + 1 = 8 Row 4: 24 = 1 + 4 + 6 + 4 + 1 = 16

Theorem 1: The sum of the numbers in row n is equal to 2n, when n is the number of the row. That đ?’? đ?‘› Ďƒ is đ?‘–=0 = 2n. đ?’Š


Row p (prime): All inner numbers are divisible by p.

Row 0: 1 Row 1: 1 + 1 Row 2: 1 + 2 + 1 Row 3: 1+3+3+1 Row 4: 1+4+6+4+1 Row 5:

1 + 5 + 10 + 10 + 5 + 1

Theorem 2: For any prime number p, all the inner numbers in row p are divisible by p.


PASCAL’S TRIANGLE in ZP (prime) Row 0 Row 3 Row 6 Row 9

1 1 1 1 2 1 1 0 0 1 1 1 0 1 1 1 2 1 1 2 1 1 0 0 2 0 0 1 1 1 0 2 2 0 1 1 1 2 1 2 1 2 1 2 1 1 0 0 0 0 0 0 0 0 1 ‌‌‌‌‌‌‌. ‌‌‌‌‌‌‌..

Pascal’s Triangle in Z3

Corollary 3: For any Pascalâ&#x20AC;&#x2122;s triangle in Zp all the inner đ?&#x2018;? n numbers of row p or p are 0â&#x20AC;&#x2122;s, meaning =0(mod p), for 0 đ?&#x2018;&#x2DC; < k < p.


Row n=2i: All inner numbers are even.

Row 0: 1 Row 1: 1+1 Row 2: 1 + 2 + 1 Row 3: 1 + 3 + 3 + 1 Row 4: 1 + 4 + 6 + 4 + 1 Row 5: 1 + 5 + 10 + 10 + 5 + 1

Theorem 4: If n=2i , then all the inner numbers in row nth are even. đ?&#x2018;&#x203A; đ?&#x2018;&#x203A; i That is if = 2 , then = 2k, for i = 1,2 ,3, â&#x20AC;Ś, n-1. đ?&#x2018;&#x2013; 1


Row n=2k-1 : All numbers are odd. Row 0: 1 Row 1: 1+1 Row 2: 1+ 2 + 1 Row 3: 1 + 3 + 3 + 1 Row 4: 1+4+6+6+4+1 Row 5: 1 + 5 + 10 + 10 + 5 + 1 Row 6: 1 + 6 + 15 + 20 + 15 + 6 + 1 Row 7: 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1

đ?&#x2018;&#x203A; Theorem 5: If the term of row = (2k â&#x20AC;&#x201C; 1) for 1 some integer k > 0 , all the numbers in the row nth are odd. 1st

nth,


PASCAL’S MATRIX

The Pascal’s Matrix is a square matrix with 1’s in the 1st column and the 1st row. Then use the Pascal’s triangle rule to get all the inner cells. That is Cell(i,j)=Cell(i-1, j)+Cell(i, j-1).

The Pascal’s triangle is the upper left of the Pascal’s matrix above.


THE DETERMINANT OF PASCAL’S MATRIX

Theorem 6: The determinant of any n x n Pascal’s matrix is 1.

Proof: Let’s reduce an n x n matrix by using a row operation for every row (except the first row) with the following rule: ri – ri-1 ri starting from row nth. Because the matrix is created using Pascal’s triangle, meaning Cell(i,j)=Cell(i-1, j)+Cell(i, j-1), it implies that Cell(i,j)-Cell(i-1, j) = Cell(i, j-1).

Since each row is shifted to the right (except the first row), all diagonal cells are 1’s and all cells in the lower left triangle are 0’s. The determinant of this upper triangular matrix is equal to 1. By properties of determinant, the determinant of n x n Pascal’s matrix is also equal to 1. Q.E.D


PATTERN FOR PASCAL’S MATRIX IN Z P For any prime p, the basic p x p Pascal’s block is of the form: i. Row 1 and Column 1 contain only “1.” ii. Row 2 and Column 2 contain all numbers from 1 to p-1 with increment 1. iii.All the cells in the lower right triangle are 0’s. (This is the result of Corollary 3.) 1

1

1

1

1

1

2

3

p-1

0

1

3

*

0

0

1

p-1

0

0

0

1

0

0

0

0


PATTERN FOR PASCAL’S MATRIX IN ZP Let A be the basic p x p Pascal’s block We then can build a bigger block in Zp, say p2 x p2 matrix from the basic block as follow: aij*A, where aij is an element of the basic block.

25x25 in Z5

Theorem 7: For any prime number p, any block in pk x pk Pascal’s matrix in Zp is a scalar product of cell(i,j) and the pk-1x pk-1 block, where cell(i,j) is from the basic block p x p.


Some pk x pk Pascal’s Matrices in Zp

The upper left triangle of of 32x32 Pascal’s matrix in Z2 is as known as Sierpinski’s Triangle


SOME P K X P K PASCAL’S MATRICES IN ZP

27x27 Pascal’s matrix in Z3

49x49 Pascal’s matrix in Z7


PATTERN FOR PASCALâ&#x20AC;&#x2122;S MATRIX IN Z N (FOR N = P K ) In Z3 (27x27)

For any pj x pj, in Zpk there is no lower-zero triangles as seen in Zp.

In Z9 (81x81)

The lower right triangle is filled with the basic matrix and smaller solid-zero triangles. The maximum size of the leg of each

pi

x

pi

k

Zp block is

đ?&#x2018;?đ?&#x2018;&#x2013; đ?&#x2018;?đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1

â&#x2C6;&#x2019; 1.


PATTE RN F O R PAS C AL’S M AT R I X I N Z N ( F O R N = 2 K)

32x32 Pascal’s matrix in Z4 32x32 Pascal’s matrix in Z2

32x32 Pascal’s matrix in Z8


Pattern For Pascal’s Matrix in Zpq (p and q are primes.) For any Pascal’s matrix in Zpq size qkxqk, the pattern for Zpq is similar to the pattern of Zp assuming that q > p. The upper half triangle (which is the Pascal’s triangle) in this matrix is similar to the upper half triangle in the same size matrix of Zq; whereas the lower half triangle of this matrix has the same shape as in the triangle in Zp.

Z6

Theorem: For any primes p and q and p < q, there is an isomorphism mapping f :ZpxZq  Zpq


Z3

Z62

27x27 matrices mapping f :Z2 x Z3 ď&#x192; Z6


The upper halfhalf triangle matrix the same The lower triangleof of Z Z66 matrix hashas the same shape the triangle triangle in in Z2.Z shape as as ininthe 3 Z23

Z6

27x27 matrices mapping f :Z2 x Z3 ď&#x192; Z6


25x25 matrices Mapping f :Z2 x Z5 ď&#x192; Z10

Z5

Z2 Z 10


(100x100) Z10

(25x25) Z10

Website: phan-yamada.weebly.com Email: tphanya3@calstatela.edu


Translating the Catholic Eucharist as the Tupinambรก Pledge of Vengeance Kittiya Lee History Department


How to teach the Eucharist to Amerindians? • Priest: That which the priest gives us in the host is God? • Neophyte: [It is] God. • P: It is not bread? • N: In no way is it bread. • P: Then what white thing is that which we see when the priest lifts up the host in Mass and when he places it in the mouth of the communicant: in no way is it bread?


• Pre-existing Tupi words • Cauim  wine • Miapé  bread

Translation Strategies, 1500s-1600s

• Portuguese loanwords • Santissimo Sacramento (“Holy Sacrament”) • Hostia (“host”) • Eucaristia (“Eucharist”)


• Pre-existing Tupi words • Cauim  wine • Miapé  bread

Translation Strategies, 1500s-1600s

• Portuguese loanwords • Santissimo Sacramento (“Holy Sacrament”) • Hostia (“host”) • Eucaristia (“Eucharist”)


Cauim feast in “America,” 1606, by Dutch cartographer Jodocus Hondius


Enemies formed the same class of beings as gods.


Portuguese loanwords

The Eucharist’s New Name (1700s)

Tupã (“God”) or Tupã pyá (“heart of God”) •


Portuguese loanwords

The Eucharist’s New Name (1700s)

Tupã (“God”) or Tupã pyá (“heart of God”) •


The eating of an enemy/divinity is the means to eternal salvation, happiness.


Tupi thunder divinity


• Eighteenth-century Christian Indians nativized the Eucharist. • They “Tupi-ized” the Eucharist.


Jê telling of

Tupi


â&#x20AC;˘ The Americas faced colonizations that were not only European.

â&#x20AC;˘ Remember the plurality of our historical and modern worlds.


Thank you!


Avian ecology in Southern California and beyond

Eric Wood Lab of Avian Ecology and Conservation www.ericmwood.org

ericmwood@calstatela.edu


The Wood lab

• Currently five MSc students – Three graduating this spring – Two more arriving in the fall

• Avian ecology, urban ecosystems, landscape and quantitative ecology, conservation


Birds in the city What is the role of the Los Angeles urban forest for sustaining biodiversity?


Birds in the city Townsendâ&#x20AC;&#x2122;s Warbler (Setophaga townsendi)

Why are migratory birds a critical component of our urban biodiversity?


Birds in the city - Why do we care? 1. Value of the LA urban forest for migratory birds? 2. Recommendations ď&#x192; wildlife and urban functioning


Birds in the city - motivations

Low income (e.g. Boyle Heights)

High income (e.g. San Marino)


Birds in the city â&#x20AC;&#x201C; research objectives 1. Where do birds hang out in the city and why?

2. Which trees do migratory birds prefer for feeding?


Birds in the city – sampling design

Median Household Income - 2014 $10,000 - $52,000 = Low Income $52,001 - $76,000 = Medium Income $76,001 - $241,528 = High Income

Sampling Design •

Socio-economic gradient (low-medium-high – depicted with light to darker blues)

45 different sampling locations (yellow circles) throughout LA (> 20,000 trees)


Birds in the city – sampling design Forest birds – 26 species (expected positive association with tree cover)

Black-throated Gray Warbler (Setophaga nigrescens) Northern Mockingbird (Mimus polyglottos)

Urban birds – 10 species (expected negative association with tree cover)


Forest bird richness

Forest bird abundance

Birds in the city â&#x20AC;&#x201C; results (obj. 1)

R2 = 0.24, p < 0.01

R2 = 0.26, p < 0.01 % tree cover

% tree cover

R2 = 0.29, p < 0.01

Urban bird richness

Urban bird abundance

R2 = 0.30, p < 0.01

% tree cover

% tree cover


Forest bird richness

Forest bird abundance

Birds in the city â&#x20AC;&#x201C; results (obj. 1)

R2 = 0.24, p < 0.01

R2 = 0.26, p < 0.01 % tree cover

% tree cover

R2 = 0.29, p < 0.01

Urban bird richness

Urban bird abundance

R2 = 0.30, p < 0.01

% tree cover

% tree cover


Birds in the city - results Urban birds here!

Forest birds here!


Birds in the city â&#x20AC;&#x201C; results (obj. 2) 100 80 %

Migratory bird tree use Street tree density

60 40

Ruby-crowned Kinglet (Regulus calendula)

20 0 Native street trees n = 229

Exotic street trees n = 2105

Migratory forest birds LOVE native trees (e.g. Coast Live Oak, California Sycamore)!!!


Birds in the city â&#x20AC;&#x201C; results (obj. 2) 100 80 %

Migratory bird tree use Street tree density

60 40

Ruby-crowned Kinglet (Regulus calendula)

20 0 Native street trees n = 229

Exotic street trees n = 2105

Migratory forest birds LOVE native trees (e.g. Coast Live Oak, California Sycamore)!!!


Take homes 1.

Urban forest critically important for migratory birds

2.

Large differences in urban canopy across socio-economic gradient â&#x20AC;&#x201C; influences bird distributions

3.

Birds love native trees! But, they also utilize exotics as well

Blue-gray Gnatcatcher (Polioptila caerulea)


Take homes 1.

Urban forest critically important for migratory birds

2.

Large differences in urban canopy across socio-economic gradient â&#x20AC;&#x201C; influences bird distributions

3.

Birds love native trees! But, they also utilize exotics as well

Blue-gray Gnatcatcher (Polioptila caerulea)


1. Big Bend NP Rio Grande Restoration

Monitoring the response of bird and butterfly communities to the removal of invasive plant species


2. Lower Owenâ&#x20AC;&#x2122;s River restoration Bringing a riparian system back to life: the response of bird and butterfly communities to the restoration of the Lower Owenâ&#x20AC;&#x2122;s River


3. Santa Monica Mountains

Effects of urbanization (fragmentation, sound and light) on birds, butterflies, herps and reptiles


La fin!

Eric Wood Lab of Avian Ecology and Conservation www.ericmwood.org

ericmwood@calstatela.edu


Identity/Politics: Intersecting Distinctions in Digital Public Spheres May 2018 - Preliminary Explorations and Evidence Dmitri Seals, Sociology dseals2@calstatela.edu


The Broader Project

1. Structural Transformations: How does polarization change political speech? 2. Intersectional Boundaries: How did boundary work in race/gender/class shift? 3. Changing Minds: How do forums influence beliefs and values?


Takeaways

1. Leaving politics: Significant drops in political discussion after 1/17, particularly on left

2. Alt-right: from political struggle to cultural subversion 3. Aligning Boundaries: anti-elite & anti-immigrant anti-woman & anti-Islam


The Broader Project

https://voat.co/v/Identitarian/1664812


Background - The Dataset 1. Huge. >1.7b comments since 2005, >1TB - reasonably mainstream since 2012 2. Good for semantic analysis. Cooccurrence matrices for comments far cleaner than for docs, which often include multiple frames 3. Structure well suited to research questions. Comment characteristics: a. ID and field position: subreddit, author, ID, b. Network structure: parent ID(to which it responds) c. Semantic analysis: comment body d. Resonance and reaction: score, moderator status, replies 4. Partisan subfields. Subset of forums: 11 left, 11 right, 6 mainstream subreddits, >5000 comments, >30 posts per month


Universe of Subreddit Networks: Shared Authors


Political Subreddit Networks: Shared Moderators


Background - Subfield Relative Size Relative size. Mainstream draws far more users than either partisan space, right is substantially larger than left. All following metrics account for relative size.


Modes of Partisan Struggle - Distinct authors v. Total comments (% of Oct 2016) Distinct authors. Authors rise through January then decline in April. Very sharp decline on left after largest rise to January, lowest decline in mainstream

Total comments. On the left and to a lesser extent on the right, authors raise while total comments drop for January, signaling shallow engagement by a large group.


Modes of Partisan Struggle - Distinct authors v. Total comments (% of Oct 2016) Distinct authors. Authors rise through January then decline in April. Very sharp decline on left after largest rise to January, lowest decline in mainstream

Total comments. On the left and to a lesser extent on the right, authors raise while total comments drop for January, signaling shallow engagement by a large group.


Modes of Partisan Struggle - Comments per author Mobilization: Partisan surge coming into elections. The surge in activity on the right was driven largely by The_Donald and altright. Communities of very active authors on the left associated with Bernie sanders steeply declined in average comments from April 2016 through January 2017. The graph at right shows the 4 subreddits achieving more than 15 comments per user over the period.


Capital distributions - Author Activity Shallow left engagement, right inequality. Though right subfields feature strong dominance by the top 1% of posters, they also include a higher proportion of active authors (>5 posts per month). Active and hyperactive (>30 posts) authors declined on the left since October.


Top phrases left v. right by distinctiveness from mainstream

Left

Rank April 16 incremental change 1 endorsed clinton 2 co-ops 3 bourgeois 4 voting green 5 progressive candidates 6 crime bill 7 clinton campaign 8 Nina (turner) 9 10 hrc

July 16 socialist alternative electoral politics ranked choice glyphosate supporting jill chris hedges that workers cooperatives gmo labeling latestagecapitalism

October 16 rojava libertarian socialist * exploitation state capitalist libertarian socialism (lesser) evilism ajamu nodapl * capitalist socialist society

January 17 * capitalisms wolff electoral politics * neoliberal corporate democrats jacobin Medicare * rojava progressive change * reactionaries

April 17 corporate democrats under capitalism ESS (Enough Sanders Spam) berniecrats tankie economic populism social democratic jacobin * neoliberals progressive candidate

Right

Rank April 16 maga * 1 high energy 2 cucks 3 immigration policy 4 illegal aliens 5 golly 6 illegal immigration 7 shitlord 8 pro trump 9 10 soros

July 16 (okie) doke no brakes (trump train) * cuck * tendies (get this patriot) a coat r/asktrumpsupporters * maga nimble navigator pepes faggot

October 16 r/redacted * cuck the 33k (emails) (red) pilling fucking white (male) * maga hiliary autists the veritas (videos) pepes

January 17 * cuck praise kek red pilled trump train 4d chess mobile link BTFO NWO tucker (carlson) * maga

April 17 shariablue libtard red pills kek virtue signal spez full retard * cuck faggot tranny


Shifting Targets of Boundary Work


Shifting Targets in Boundary Work


Realignments in Political Affiliation & Boundary Work


Realignments in Boundary Work


Takeaways

1. Leaving politics: Significant drops in political discussion after 1/17, particularly on left

2. Alt-right: from political struggle to cultural subversion 3. Aligning Boundaries: anti-elite & anti-immigrant anti-woman & anti-Islam


Questions for the Future

1. Structural Transformations: What has polarized, who speaks where, who drops out? 2. Intersectional Boundaries: How will boundary work in race/gender/class continue to shift? 3. Changing Minds: How does prolonged exposure to a field change behavior?


Identity/Politics: Intersecting Distinctions in Digital Public Spheres

Thanks for your time! Dmitri Seals, Sociology dseals2@calstatela.edu


Hydrologic Research in Urban California Coastal Watersheds by Barry Hibbs California State University, Los Angeles


Everybodyâ&#x20AC;&#x2122;s Image of LA Urban Channels


THEM!

(1954)


The Gumball Rally (1976)

Point Blank (1967)


Terminator 2

(1991)

The Gang Movie Genre


How and why do you do research in urban concrete lined channels?


Objectives •

Summarize a case study investigation in an urban channel that provides new conceptual models of stream/aquifer interactions and trace element loading to a protected watershed and estuary

Describe management implications of lined and unlined channels in urban catchments and describe water quality implications for restoration

All work presented here applies to gaining stream conditions during dry weather flows


Objective 1: Case Study Eye-Opening Hydrochemical Study El Modena Concrete-Lined Channel


El Modena Channel Groundwater Swamp of Frogs

% alkali


SeO42SO42-

Mountain Runoff

Selenium and Sulfur Sequestered in Sediments

Selenium and Sulfur Oxidized in Sediments

SeO42SO42-


Objective 2: Management Implications of Lined and Unlined Channels in Urban Catchments - Water Quality Implications


Biogeochemical Processes in Unlined Channels

Floodplain

Creek water table Vegetation uptake

Hillslope Bedrock

Nitrate reduction Trace element reduction/uptake

Precipitation Recharge Groundwater Flow

Organic contaminant removal


Lined Channels Suppress Biogeochemical Processes

Floodplain Grout Lined Channel

water table

Hillslope Bedrock Precipitation Recharge

Groundwater Flow


Grout Lined Channel Groundwater

Grout

Surfacewater

seepage Oxidizing

Oxidizing

No intrinsic removal (e.g., no nitrate reduction, no trace element uptake)

Natural or Restored Channel Groundwater

Vegetated Streambed

Surfacewater

seepage Oxidizing

Reducing

Oxidizing

Intrinsic bioremediation/removal (e.g., denitrification, trace element uptake)


Natural or Restored Channel Groundwater

Vegetated Streambed

Surfacewater

seepage Oxidizing

Reducing

Oxidizing

Intrinsic bioremediation/removal (e.g., denitrification, trace element uptake)

Denitrification 4NO3- + 5CH2O â&#x2020;&#x2019; 2N2 + 5HCO3- + H+ + 2H2O Nitrate

Organic Matter

N-Gas

Bicarbonate

H-Ion

Selenium Reduction SeO42- + 2C-organic + 2H2O â&#x2020;&#x2019; H2Se + 2HCO3Selenate

Organic Matter

Selenide gas Bicarbonate


Products of Our Research in Urban Watersheds 26 journal papers and book chapters 79 published abstracts

16 contract reports 21 MS Theses completed

Two MS Theses underway (Marissa DeHoyos; Geoffrey Clifford) Funding: NSF, NOAA, UC-Riverside, Cal State Water Board, US National Park Service


Ending


THE NEAR-PEER ADVANTAGE Chris Endy, Birte Pfleger, Carole Srole


F L IPPIN G , PEER M EN TO R ING AN D TEAM - BASED L EAR N IN G

This flowchart shows how our redesign uses three main elements (flipping, peer mentoring, and scaffolding) to foster improved student academic skills and motivation. For more details, see our forthcoming SoTL publication: Carole Srole, Christopher Endy, and Birte Pfleger, “Active Learning in History Survey Courses: The Value of In-Class Peer Mentoring,” History Teacher 51 (December 2017): 89-102.


- H I G H E R PA S S R AT E S

• DFW Rates in GE US History Survey Classes

-IMPROVED WRITING & HISTORICAL THINKING -INCREASED STUDENT ENGAGEMENT

HIST 2010 students completing a Team Quiz, Fall 2017

Grade data from Pfleger’s Spring 2017 large lecture class


PEER MEN TO R ING AS A H IGH -IMPACT L EAD ER SH IP EXPER IEN C E

A peer mentor’s reflection at the end of the course:

• "This experience reinforced my aspirations of becoming a professor one day.… I did not expect the class to turn out as well as it did, as I was very nervous about what I was getting myself into…. in the end it was one of the most memorable, enjoyable experiences I have had in my collegiate career.” • A former peer mentor’s reflection after teaching for the first time at a nearby community college:

• “I really could not have done this without your assistance. I borrowed so many parts of History 202A and when I had to improvise I found myself thinking ‘What would Dr. Pfleger do?’” Peer Mentors for HIST 2010 in Fall 2017


ASSIGNMENT DESIGN CAN MAKE SCIENCE TRANSPARENT Kirsten Fisher, Biological Sciences


• The Transparency in Learning and Teaching in Higher Education project (TILT Higher Ed)

TRANSPARENT TEACHING

• Basic premise: making the purpose, task, and criteria for an assignment explicit to students • Transparent assignment design template • Easy to implement

Dr. Mary-Ann Winkelmes, UNLV


ASSIGNMENT REDESIGN


REFLECTION…

• Is figuring out the assignment and/or criteria for success the (implicit) learning outcome? • Can that skill be exposed and developed incrementally through increasingly independent assignments? • …For students to learn to think like scientists, it might help if the scientist teaching them explained their purpose and rationale for assignments

Lightning Round - Research/Teaching Presentations  
Lightning Round - Research/Teaching Presentations  
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