Qwen2-VL: To See the World More Clearly

The image shows a stack of colorful blocks with numbers on them. Here is the color and number of each block from top to bottom:

• Top row (single block): Blue with the number 9
• Second row (two blocks): Light blue with the number 7, and green with the number 8
• Third row (three blocks): Purple with the number 4, pink with the number 5, and light green with the number 6
• Bottom row (four blocks): Red with the number 0, orange with the number 1, yellow with the number 2, and light green with the number 3

1. If $i > 0$, let $\underline{d}’’ := (i-1, d-i-l+1, l)$. Let $V_{\underline{d}}$ and $V_{\underline{d}’’}$ be $r+1$-dimensional subspaces of $H^0(\mathcal{L}_{\underline{d}})$ and $H^0(\mathcal{L}_{\underline{d}’’})$, respectively, such that $\varphi_{\underline{d}, \underline{d}’’}(V_{\underline{d}}) = V_{\underline{d}’’}^{X_1, 0}$. Then the distributivity holds in $V_{\underline{d}’’}$ if and only if $$ \dim (V_{\underline{d}}^{X_2, 0} + V_{\underline{d}}^{X_3, 0}) - \dim (V_{\underline{d}’’}^{X_2, 0} + V_{\underline{d}’’}^{X_3, 0}) = r + 1 - \dim (V_{\underline{d}’’}^{X_1, 0} + V_{\underline{d}’’}^{X_2, 0} + V_{\underline{d}’’}^{X_3, 0}) $$

2. If $i + l \leq d - 1$, let $\underline{d}’’’ := (i, d-i-l-1, l+1)$. Let $V_{\underline{d}}$ and $V_{\underline{d}’’’}$ be $r+1$-dimensional subspaces of $H^0(\mathcal{L}_{\underline{d}})$ and $H^0(\mathcal{L}_{\underline{d}’’’})$, respectively, such that $\varphi_{\underline{d}’’’, \underline{d}}(V_{\underline{d}’’’}) = V_{\underline{d}}^{X_3, 0}$. Then the distributivity holds in $V_{\underline{d}}$ if and only if $$ \dim (V_{\underline{d}’’’}^{X_1, 0} + V_{\underline{d}’’’}^{X_2, 0}) - \dim (V_{\underline{d}}^{X_1, 0} + V_{\underline{d}}^{X_2, 0}) = r + 1 - \dim (V_{\underline{d}}^{X_1, 0} + V_{\underline{d}}^{X_2, 0} + V_{\underline{d}}^{X_3, 0}), $$ where the maps $\varphi_{\underline{d}, \underline{d}’’}$ and $\varphi_{\underline{d}’’’, \underline{d}}$ in Proposition 3.14 are the maps linking the corresponding sheaves. Another important result is Proposition 3.16, which establishes an inequality for any exact limit linear series. Specifically, our Proposition 3.16 says:

Let ${(\mathcal{L}_{\underline{d}}, V_{\underline{d}})}_{\underline{d}}$ be an exact limit linear series of degree $d$ and dimension $r$. Then $$ \sum_{\underline{d}} \dim \left( \frac{V_{\underline{d}}}{V_{\underline{d}}^{X_1, 0} + V_{\underline{d}}^{X_2, 0} + V_{\underline{d}}^{X_3, 0}} \right) \geq r + 1. $$

As a consequence of Proposition 3.14 and Proposition 3.16, in Corollary 3.17, we get the following characterization of exact limit linear series satisfying the distributivity at each multidegree:

Let ${(\mathcal{L}_{\underline{d}}, V_{\underline{d}})}_{\underline{d}}$ be an exact limit linear series of degree $d$ and dimension $r$. Then $$ \sum_{\underline{d}} \dim \left( \frac{V_{\underline{d}}}{V_{\underline{d}}^{X_1, 0} + V_{\underline{d}}^{X_2, 0} + V_{\underline{d}}^{X_3, 0}} \right) = r + 1 $$ if and only if the distributivity holds in $V_{\underline{d}}$ for any $\underline{d}$.

In Section 3, we also study simple limit linear series. In Proposition 3.18, we characterize the multidegrees which determine all simple bases. In Proposition 3.20, following the construction in Muñoz [5], Proposition 4.4, we present a certain structure of linearly independent sections for any exact limit linear series satisfying the distributivity property. In propositions 3.21 and 3.22, we obtain some properties of that structure. These properties imply the existence of a simple basis (we give a proof of that in Proposition 3.23).

We conclude Section 3 with a characterization of simple limit linear series among the exact limit linear series. Specifically, our Corollary 3.24 says:

Let ${(\mathcal{L}_{\underline{d}}, V_{\underline{d}})}_{\underline{d}}$ be an exact limit linear series of degree $d$ and dimension $r$. Then the following statements are equivalent:

1. ${(\mathcal{L}_{\underline{d}}, V_{\underline{d}})}_{\underline{d}}$ is simple.
2. $\sum_{\underline{d}} \dim \left( \frac{V_{\underline{d}}}{V_{\underline{d}}^{X_1, 0} + V_{\underline{d}}^{X_2, 0} + V_{\underline{d}}^{X_3, 0}} \right) = r + 1$.
3. The distributivity holds in $V_{\underline{d}}$ for any $\underline{d}$.

In a subsequent work, given a refined limit linear series, we will construct all simple extensions by combining our Proposition 3.14 and the method of Muñoz [6] for the construction of all exact extensions.

1. 汉语，也称为“华语”。是中国使用人数最多的语言，也是世界上作为第一语言使用人数最多的语言。是中华优秀传统文化的重要载体。繁体字，又稱為“繁體中文”，與“簡化字”/“簡體字”相對。一般是指漢字簡化運動被簡化字所代替的漢字。

   • 语言：中文

2. 日本語は地方ごとに多様な方言があり、とりわけ琉球諸島で方言差が著しい。

   • 语言：日语

3. 한국어(韓國語), 조선말(朝鮮말)는 대한민국과 조선민주주의인민공화국의 공용어이다. 둘은 표기나 문법에서는 차이가 없지만 동사 어미나 표현에서 차이가 있다.

   • 语言：韩语

4. Le français est une langue indo-européenne de la famille des langues romanes dont les locuteurs sont appelés « francophones ».

   • 语言：法语

5. El español o castellano es una lengua romance procedente del latín hablado, perteneciente a la familia de lenguas indoeuropeas.

   • 语言：西班牙语

6. A língua portuguesa, também designada português, é uma língua indo-europeia românica flexiva ocidental originada no galego-português falado no Reino da Galiza e no norte de Portugal.

   • 语言：葡萄牙语

7. Is ceann de na teangacha Ceilteacha í an Ghaeilge (nó Gaeilge na hÉireann mar a thugtar uirthi corruair), agus ceann de na trí cinn de theangacha Ceilteacha ar a dtugtar na teangacha Gaelacha (Gaeilge, Gaeilge Mhanann agus Gaeilge na hAlban) go háirithe.

   • 语言：爱尔兰语

8. English is a West Germanic language in the Indo-European language family, whose speakers, called Anglophones, originated in early medieval England on the island of Great Britain.

   • 语言：英语

9. Die deutsche Sprache bzw. Deutsch ist eine westgermanische Sprache, die weltweit etwa 90 bis 105 Millionen Menschen als Muttersprache und weiteren rund 80 Millionen als Zweit- oder Fremdsprache dient.

   • 语言：德语

10. Język polski, polszczyzna — język z grupy zachodniosłowiańskiej (do której należą również czeski, kaszubski, słowacki, języki łużyckie czy wymarły język drzewiański), stanowiącej część rodziny indoeuropejskiej.

    • 语言：波兰语

11. Η ελληνική γλώσσα ανήκει στην ινδοευρωπαϊκή οικογένεια και αποτελεί το μοναδικό μέλος του ελληνικού κλάδου.

    • 语言：希腊语

12. Tiếng Việt hay Việt ngữ là một ngôn ngữ thuộc ngữ hệ Nam Á, được công nhận là ngôn ngữ chính thức tại Việt Nam.

    • 语言：越南语

13. Монгол хэл нь Монгол улсын албан ёсны хэл юм.

    • 语言：蒙古语

14. Число владеющих русским языком в России составляет 137,5 млн человек (2010). Всего в мире на русском говорят 258,2 млн человек (2022)

    • 语言：俄语

15. हिन्दी विश्व की एक प्रमुख भाषा है और भारत की एक राजभाषा है। केन्द्रीय स्तर पर भारत में सह-आधिकारिक भाषा अंग्रेज़ी है।

    • 语言：印地语

16. Kiswahili ni lugha ya Kibantu yenye misamiati mingi ya Kiarabu (35%), lakini sasa ya Kiingereza pia (10%), inayozungumzwa katika eneo kubwa la Afrika ya Mashariki.

    • 语言：斯瓦希里语

尊敬的老师

展信舒颜

气温骤降

树叶无声无息的掉落

教诲如春风

师恩深似海

愿温馨、喜悦与幸福

如同吐露芬芳的花朵

永远洋溢在您幸福的时光里

To solve the “Snake in Matrix” problem, we need to implement a function that takes the size of the matrix n and a list of commands, and returns the final position of the snake after executing all the commands. Here’s a step-by-step approach to solve this problem:

1. Initialize Variables:

   • Initialize the current position of the snake as 0.
   • Initialize the direction of the snake as “RIGHT” (initially, the snake moves to the right).

2. Define Movement Functions:

   • Create a function to move the snake based on the current direction and the size of the matrix n.
   • Update the position of the snake accordingly.

3. Process Commands:

   • Iterate through each command in the list of commands.
   • For each command, update the direction and position of the snake.

4. Return Final Position:

   • After processing all commands, return the final position of the snake.

Here’s the Python code to implement the solution:

class Solution:
    def finalPositionOfSnake(self, n: int, commands: List[str]) -> int:
        # Initialize variables
        position = 0
        direction = "RIGHT"
        
        # Define movement functions
        def move_right():
            nonlocal position
            position += 1
        
        def move_down():
            nonlocal position
            position += n
        
        def move_left():
            nonlocal position
            position -= 1
        
        def move_up():
            nonlocal position
            position -= n
        
        # Process commands
        for command in commands:
            if command == "RIGHT":
                move_right()
            elif command == "DOWN":
                move_down()
            elif command == "LEFT":
                move_left()
            elif command == "UP":
                move_up()
        
        # Return final position
        return position

This code defines a class Solution with a method finalPositionOfSnake that takes the size of the matrix n and a list of commands, and returns the final position of the snake after executing all the commands. The movement functions (move_right, move_down, move_left, move_up) are used to update the position of the snake based on the current direction.

Since triangle ABC is isosceles, we know that AC = BC. Therefore, we can set the expressions for AC and BC equal to each other and solve for x:

4x + 1 = 5x - 0.5

Subtracting 4x from both sides, we get:

1 = x - 0.5

Adding 0.5 to both sides, we get:

1.5 = x

Now that we know the value of x, we can substitute it back into the expression for AC to find its length:

AC = 4x + 1

AC = 4(1.5) + 1

AC = 6 + 1

AC = 7

Therefore, the length of AC is 7.

The page titles in the screenshot of the search result are:

1. 你好，Qwen2
2. QwenLM/Qwen2
3. Qwen2 - a Qwen Collection
4. 快速开始
5. Qwen2-7B
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9. [2407.10671] Qwen2 Technical Report
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To determine the number of days between Ruka’s and Yuki’s birthdays, we need to calculate the difference between their respective birth dates.

1. Ruka’s birthday is on June 24.
2. Yuki’s birthday is on September 17.

Let’s break it down step by step:

1. From June 24 to June 30:

   • June 24 to June 30 is 6 days.

2. From July 1 to August 31:

   • July has 31 days.
   • August has 31 days.
   • Total days in July and August: 31 + 31 = 62 days.

3. From September 1 to September 17:

   • September 1 to September 17 is 17 days.

Now, we add all these days together:

• June 24 to June 30: 6 days
• July 1 to August 31: 62 days
• September 1 to September 17: 17 days

Total days = 6 + 62 + 17 = 85 days

Therefore, there are 85 days between Ruka’s and Yuki’s birthdays.

以下是图中每日的最高气温和最低气温整理成的表格：

希望这对你有帮助！