The flowers in the picture are forget-me-nots (Myosotis). These small, delicate flowers are known for their vibrant colors, typically blue, pink, or white, and are often used in bouquets and gardens for their charming appearance.
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
$$
\dim (V_{\underline{d}}^{X_2, 0} + V_{\underline{d}}^{X_3, 0}) - \dim (V_{\underline{d}’’}^{X_2, 0} + V_{\underline{d}’’}^{X_3, 0}) = r + 1 - \dim (V_{\underline{d}’’}^{X_1, 0} + V_{\underline{d}’’}^{X_2, 0} + V_{\underline{d}’’}^{X_3, 0})
$$
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:
${(\mathcal{L}_{\underline{d}}, V_{\underline{d}})}_{\underline{d}}$ is simple.
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.
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
Le français est une langue indo-européenne de la famille des langues romanes dont les locuteurs sont appelés « francophones ».
语言：法语
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
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
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
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
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
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
Η ελληνική γλώσσα ανήκει στην ινδοευρωπαϊκή οικογένεια και αποτελεί το μοναδικό μέλος του ελληνικού κλάδου.
语言：希腊语
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
Монгол хэл нь Монгол улсын албан ёсны хэл юм.
语言：蒙古语
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
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
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.
The text on the wall is in Portuguese and reads: “não é normal ter medo de andar sozinha na rua!!!” This translates to: “It is not normal to be afraid to walk alone on the street!!!”
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:
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).
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.
Process Commands:
Iterate through each command in the list of commands.
For each command, update the direction and position of the snake.
Return Final Position:
After processing all commands, return the final position of the snake.
Here’s the Python code to implement the solution:
classSolution:deffinalPositionOfSnake(self,n:int,commands:List[str])->int:# Initialize variablesposition=0direction="RIGHT"# Define movement functionsdefmove_right():nonlocalpositionposition+=1defmove_down():nonlocalpositionposition+=ndefmove_left():nonlocalpositionposition-=1defmove_up():nonlocalpositionposition-=n# Process commandsforcommandincommands:ifcommand=="RIGHT":move_right()elifcommand=="DOWN":move_down()elifcommand=="LEFT":move_left()elifcommand=="UP":move_up()# Return final positionreturnposition
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.
The video begins with a man speaking to the camera, followed by a group of people sitting in a control room. The camera then cuts to two men floating inside a space station, where they are seen speaking to the camera. The men appear to be astronauts, and they are wearing space suits. The space station is filled with various equipment and machinery, and the camera pans around to show the different areas of the station. The men continue to speak to the camera, and they appear to be discussing their mission and the various tasks they are performing. Overall, the video provides a fascinating glimpse into the world of space exploration and the daily lives of astronauts.
User
What is the color of the clothes the two astronauts wearing?
Qwen2-VL
The two astronauts are wearing blue and black clothes.
如需使用 Qwen2-VL 系列的最强模型 Qwen2-VL-72B，目前您可以通过我们的官方 API （登录 DashScope 注册并获取API Key）来体验该模型：
fromopenaiimportOpenAIimportosimportbase64defencode_image(image_path):withopen(image_path,"rb")asimage_file:returnbase64.b64encode(image_file.read()).decode("utf-8")# Path to your imageimage_path="dog_and_girl.jpeg"# Getting the base64 stringbase64_image=encode_image(image_path)defget_response():client=OpenAI(api_key=os.getenv("DASHSCOPE_API_KEY"),base_url="https://dashscope.aliyuncs.com/compatible-mode/v1",)completion=client.chat.completions.create(model="qwen-vl-max-0809",messages=[{"role":"user","content":[{"type":"text","text":"What is this?"},{"type":"image_url","image_url":{"url":"https://dashscope.oss-cn-beijing.aliyuncs.com/images/dog_and_girl.jpeg"},},{"type":"image_url","image_url":{"url":f"data:image/jpeg;base64,{base64_image}"},},],}],top_p=0.8,stream=True,stream_options={"include_usage":True},)forchunkincompletion:print(chunk.model_dump_json())if__name__=="__main__":get_response()
fromtransformersimportQwen2VLForConditionalGeneration,AutoTokenizer,AutoProcessorfromqwen_vl_utilsimportprocess_vision_info# default: Load the model on the available device(s)model=Qwen2VLForConditionalGeneration.from_pretrained("Qwen/Qwen2-VL-7B-Instruct",device_map="auto")# We recommend enabling flash_attention_2 for better acceleration and memory saving, especially in multi-image and video scenarios.# model = Qwen2VLForConditionalGeneration.from_pretrained(# "Qwen/Qwen2-VL-7B-Instruct",# torch_dtype=torch.bfloat16,# attn_implementation="flash_attention_2",# device_map="auto",# )# default processerprocessor=AutoProcessor.from_pretrained("Qwen/Qwen2-VL-7B-Instruct")# The default range for the number of visual tokens per image in the model is 4-16384. You can set min_pixels and max_pixels according to your needs, such as a token count range of 256-1280, to balance speed and memory usage.# min_pixels = 256*28*28# max_pixels = 1280*28*28# processor = AutoProcessor.from_pretrained("Qwen/Qwen2-VL-7B-Instruct", min_pixels=min_pixels, max_pixels=max_pixels)messages=[{"role":"user","content":[{"type":"image","image":"https://qianwen-res.oss-cn-beijing.aliyuncs.com/Qwen-VL/assets/demo.jpeg",},{"type":"text","text":"Describe this image."},],}]# Preparation for inferencetext=processor.apply_chat_template(messages,tokenize=False,add_generation_prompt=True)image_inputs,video_inputs=process_vision_info(messages)inputs=processor(text=[text],images=image_inputs,videos=video_inputs,padding=True,return_tensors="pt",)# Inference: Generation of the outputgenerated_ids=model.generate(**inputs,max_new_tokens=128)generated_ids_trimmed=[out_ids[len(in_ids):]forin_ids,out_idsinzip(inputs.input_ids,generated_ids)]output_text=processor.batch_decode(generated_ids_trimmed,skip_special_tokens=True,clean_up_tokenization_spaces=False)print(output_text)
