Model: cristiano-sartori/bge_ft_128
Author: cristiano-sartori
Pipeline: sentence-similarity
Library: sentence-transformers
Tags: sentence-transformers, safetensors, bert, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:1140, loss:MatryoshkaLoss, loss:MultipleNegativesRankingLoss, en, arxiv:1908.10084, arxiv:2205.13147, arxiv:1705.00652, base_model:BAAI/bge-base-en-v1.5, base_model:finetune:BAAI/bge-base-en-v1.5, license:apache-2.0, model-index, text-embeddings-inference, endpoints_compatible, region:us
Base models (finetune): id: BAAI/bge-base-en-v1.5
Card metadata: language: en
license: apache-2.0
tags: sentence-transformers, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:1140, loss:MatryoshkaLoss, loss:MultipleNegativesRankingLoss
base_model: BAAI/bge-base-en-v1.5
widget: source_sentence: Can we devise a Best-effort Broadcast algorithm that satisfies the causal delivery property, without being a causal broadcast algorithm, i.e., without satisfying the agreement property of a reliable broadcast?
sentences: SCRATCHPAD:
To determine the output of the HMM Part-of-Speech (PoS) tagger, we need to compute the probabilities of each possible tag sequence given the observed words. This process involves understanding three key components: emission probabilities, transition probabilities, and initial probabilities.

1. **Emission Probabilities**: These are the probabilities of observing a specific word given a particular tag. For example:
   - P("a" | DET) = 0.1
   - P("accurately" | ADV) = 0.1
   - P("computer" | N) = 0.1
   - P("process" | N) = 0.095
   - P("programs" | N) = 0.080

2. **Transition Probabilities**: This matrix provides the probabilities of transitioning from one tag to another. For instance:
   - P(N | DET) = 0.55, meaning if the previous tag is a determiner (DET), the next tag is likely to be a noun (N) with a probability of 55%.

3. **Initial Probabilities**: These reflect the likelihood of starting with each tag:
   - P(DET) = 0.20
   - P(N) = 0.06
   - P(V) = 0.08
   - P(ADV) = 0.07
   - P(ADJ) = 0.02

To tag a sentence using the HMM PoS tagger, we typically implement the Viterbi algorithm, which computes the most likely sequence of tags by iteratively multiplying the initial probabilities with the respective transition and emission probabilities for each word in the sentence.

Assuming the sequence of words is: "a computer process programs accurately", we can analyze the potential tags:
- "a" is most likely a DET.
- "computer" is likely a N.
- "process" can be either N or V, but given its higher emission probability as a noun, we tag it as N.
- "programs" similarly has a higher probability of being tagged as N.
- "accurately" is most likely tagged as ADV.

Utilizing the Viterbi algorithm, we can determine that the most probable sequence of tags for the words in this sentence is:
- "a" - DET
- "computer" - N
- "process" - N
- "programs" - N
- "accurately" - ADV

This output accurately reflects the tagging that maximizes the joint probability of the observed words with their corresponding tags.

FINAL_JSON:
{"answer":"The output tags are: a - DET, computer - N, process - N, programs - N, accurately - ADV."}, Implementing daily Scrum meetings can be a valuable practice for our team, especially when discussing complex features like the QR code scanning functionality for discounts. Here are my thoughts on this suggestion:

1. **Focus on Collaboration**: Daily Scrum meetings provide an excellent platform for team members to share updates on their tasks, discuss roadblocks, and collaborate on solutions. This is particularly important for a complex feature like QR code scanning, where multiple team members may need to contribute their expertise.

2. **Encouraging Communication**: By organizing these meetings, we can ensure that everyone is aligned on the goals and progress of the project. It allows for real-time feedback and ideas to flow freely, which can lead to innovative solutions for implementing the QR code feature.

3. **Addressing Complexity**: Since the QR code scanning feature may involve various considerationsâlike camera integration, backend validation, and user experienceâitâs essential for the team to discuss these aspects regularly. Daily meetings can help break down the implementation into manageable tasks and ensure that everyone is on the same page.

4. **Adapting to Challenges**: If any technical challenges arise during the implementation of the QR code feature, daily Scrum meetings provide an opportunity to address them quickly. This can help prevent bottlenecks and keep the project on schedule.

5. **Commitment to Iteration**: Scrum emphasizes iterative development, and daily meetings align well with this principle. We can assess our progress and adapt our approach to the QR code feature based on team input and testing outcomes.

In conclusion, I fully support the idea of organizing daily Scrum meetings. They will enhance our team's communication and collaboration, especially for complex features like QR code scanning, ultimately contributing to the successful development of our mobile app. Letâs ensure we keep these meetings focused and productive to maximize their effectiveness!, No, it is not feasible to create a Best-effort Broadcast algorithm that satisfies the causal delivery property without being classified as a causal broadcast algorithm. The underlying issue lies in the requirement for agreement among all participating nodes. A causal broadcast inherently includes conditions where every participant must reach consensus on message deliveries, which conflicts with the flexible nature of best-effort approaches. 

In practice, any attempt to implement a Best-effort Broadcast would fail to maintain causal relationships due to the lack of necessary synchronization mechanisms like acknowledgment protocols or timestamps. Therefore, without these elements, it becomes impossible to ensure that causality is respected among messages delivered across different nodes. Overall, the absence of an agreement property directly undermines any potential for achieving valid causal delivery in this context., source_sentence: Many general evaluation metrics can be considered for various NLP tasks. The simplest one is accuracy. Give several examples of NLP tasks for which accuracy can be used as an evaluation metric. Justify why. In general, what property(ies) must an NLP task satisfy in order to be evaluable through accuracy?
sentences: Accuracy is a straightforward evaluation metric that measures the proportion of correct predictions made by a model out of the total predictions. It is particularly suitable for tasks where the outputs are discrete classes and there is a clear correct answer for each input. Here are several examples of NLP tasks where accuracy can be effectively used as an evaluation metric:

1. **Text Classification**: In tasks such as sentiment analysis (classifying text as positive, negative, or neutral), accuracy can be used to measure how many texts were correctly classified compared to the total number of texts. Each text has a definitive label, making accuracy a suitable metric.

2. **Named Entity Recognition (NER)**: In NER, the goal is to identify and classify named entities in text (e.g., person names, organizations, locations). Accuracy can be used to evaluate how many entities were correctly identified and classified compared to the total number of entities present in the dataset.

3. **Part-of-Speech Tagging**: This task involves assigning parts of speech (e.g., noun, verb, adjective) to each word in a sentence. Accuracy is appropriate here as each word has a specific correct tag, allowing for a straightforward comparison of predicted tags against true tags.

4. **Spam Detection**: In this binary classification task, emails are classified as either "spam" or "not spam." Accuracy can be used to assess the proportion of emails that were correctly classified.

5. **Language Identification**: This task involves determining the language of a given text. Since each text belongs to a specific language, accuracy can be used to evaluate the model's performance in correctly identifying the language.

### Properties for Evaluability through Accuracy

For an NLP task to be evaluable through accuracy, it must satisfy the following properties:

1. **Discrete Output Classes**: The task must involve a finite set of discrete classes or labels. Each input should correspond to one specific output class.

2. **Clear Correctness Criteria**: There must be a clear definition of what constitutes a correct prediction. Each input should have a definitive ground truth label that can be compared against the model's prediction.

3. **Balanced Class Distribution (optional but beneficial)**: While not strictly necessary, having a balanced distribution of classes can enhance the interpretability of accuracy as a metric. In cases of class imbalance, accuracy may be misleading, as a model could achieve high accuracy by favoring the majority class.

In summary, accuracy is a suitable evaluation metric for NLP tasks that involve discrete classifications with clear correctness criteria, allowing for straightforward comparisons between predicted and true labels., It is generally not a good idea to immediately fix an unrelated bug without understanding its source and frequency, as it may divert attention from the current sprint goals and potentially introduce new issues., To solve the problem, we need to understand how smoothed probabilistic retrieval works and how document ranking is affected by the addition of new documents.

### Step 1: Understanding the Current Document Collection

We have two documents:
- \( d_1 = \text{aabc} \) 
- \( d_2 = \text{abc} \)

#### Vocabulary
The vocabulary \( V = \{ a, b, c \} \).

#### Term Frequencies
- For \( d_1 \):
  - \( f(a) = 2 \)
  - \( f(b) = 1 \)
  - \( f(c) = 1 \)
  
- For \( d_2 \):
  - \( f(a) = 1 \)
  - \( f(b) = 1 \)
  - \( f(c) = 1 \)

### Step 2: Calculate the Document Lengths
- Length of \( d_1 \): \( |d_1| = 4 \)
- Length of \( d_2 \): \( |d_2| = 3 \)

### Step 3: Applying Smoothed Probabilistic Retrieval
Using the formula for smoothed probabilities:

\[
P(w|d) = \frac{f(w, d) + \lambda \cdot P(w|C)}{|d| + \lambda \cdot |V|}
\]

where:
- \( f(w, d) \) is the term frequency of word \( w \) in document \( d \),
- \( P(w|C) \) is the probability of word \( w \) in the entire collection,
- \( |d| \) is the length of document \( d \),
- \( |V| \) is the size of the vocabulary.

### Step 4: Calculate Collection Probabilities
To calculate \( P(w|C) \), we first find the total counts:
- Total occurrences of each word in the collection:
  - \( f(a) = 2 + 1 = 3 \)
  - \( f(b) = 1 + 1 = 2 \)
  - \( f(c) = 1 + 1 = 2 \)

Total term count in the collection:
- Total terms: \( 4 + 3 = 7 \)

Now we can calculate:

\[
P(a|C) = \frac{3}{7}, P(b|C) = \frac{2}{7}, P(c|C) = \frac{2}{7}
\]

### Step 5: Calculate Probabilities for Documents
#### For \( d_1 \):
- For \( q = ab \):
  
\[
P(a|d_1) = \frac{2 + 0.5 * (3/7)}{4 + 0.5 * 3} = \frac{2 + 0.2143}{4 + 1.5} = \frac{2.2143}{5.5} ≈ 0.4026
\]

\[
P(b|d_1) = \frac{1 + 0.5 * (2/7)}{4 + 0.5 * 3} = \frac{1 + 0.1429}{5.5} ≈ 0.2070
\]

Calculating the score for \( d_1 \):
\[
Score(d_1, q) ≈ P(a|d_1) * P(b|d_1) ≈ 0.4026 * 0.2070 ≈ 0.0834
\]

#### For \( d_2 \):
- For \( q = ab \):

\[
P(a|d_2) = \frac{1 + 0.5 * (3/7)}{3 + 0.5 * 3} = \frac{1 + 0.2143}{4.5} ≈ 0.2743
\]

\[
P(b|d_2) = \frac{1 + 0.5 * (2/7)}{3 + 0.5 * 3} = \frac{1 + 0.1429}{4.5} ≈ 0.2522
\]

Calculating the score for \( d_2 \):
\[
Score(d_2, q) ≈ P(a|d_2) * P(b|d_2) ≈ 0.2743 * 0.2522 ≈ 0.0691
\]

### Step 6: Current Ranking
From the calculations:
- Score for \( d_1 ≈ 0.0834 \)
- Score for \( d_2 ≈ 0.0691 \)

So currently, we have:
\[ d_1 > d_2 \]

### Step 7: Can We Enforce Both Rankings?
To enforce both rankings, we need to find additional documents to add to the collection.

If we add documents that contain specific combinations of terms, we can adjust the scores.

#### Example Documents
If we add:
- Document \( d_3 = aaacb \)
- Document \( d_4 = bbcac \)

These documents would increase the counts of specific terms and affect both documents' probabilities.

### Conclusion
It is indeed possible to manipulate the document scores by adding suitable documents to the collection to achieve both rankings, but it would require careful selection of documents that significantly influence term probabilities.

Thus, it is possible to enforce both rankings by adding documents that adjust term frequencies appropriately, enabling us to create situations where either document can rank higher depending on the context provided by additional data in the collection., source_sentence: Assume you are writing server-side code for an online shop. The code will be invoked over HTTP from a mobile app. Your current code is as follows:
public class ShoppingCart {
    public void buy(Product product, int quantity) {
        if (product == null) { throw new IllegalArgumentException("product cannot be null"); }
        if (quantity < 1) { throw new IllegalArgumentException("quantity must be at least 1"); }

        int price = product.getUnitPrice() * quantity;
        int discount = computeDiscount(product, quantity);
        int shippingFees = computeShippingFees(product, quantity);

        int totalPrice = price - discount + shippingFees;
        // this triggers a call to the actual credit card processor
        CreditCardProcessor.billCurrentUser(totalPrice);
    }

    private int computeDiscount(Product product, int quantity) {
        // ... discount computation logic ...
    }

    private int computeShippingFees(Product product, int quantity) {
        // ... shipping fees computation logic ...
    }
}

A colleague remarks that hardcoding "CreditCardProcessor" is not a good practice, and that "ShoppingCart" should instead have a payment processor interface as a constructor parameter. Explain in 1 sentence whether this is a good idea and why or why not:
sentences: **

Yes, using a payment processor interface as a constructor parameter in the `ShoppingCart` class is a good idea because:

1. **Easier Testing**: This approach allows for the injection of mock payment processors during unit tests, enabling more effective and isolated testing of the `ShoppingCart` functionality.
2. **Loose Coupling**: By relying on an interface rather than a specific implementation, the `ShoppingCart` class becomes less dependent on the payment processor, making it easier to swap out or update payment methods in the future.
3. **Improved Maintainability**: Changes to the payment processing logic can be made independently of the `ShoppingCart` code, reducing the risk of introducing bugs and simplifying code maintenance.
4. **Adherence to Design Principles**: This design aligns with several software engineering principles, such as the Single Responsibility Principle (SRP) and the Open/Closed Principle, promoting cleaner and more modular code.

**, To show that any extreme point \( x^* \) of the linear program has at most \( m \) non-zero entries, we start by considering the constraints given by the linear program:

1. \( Ax = b \)
2. \( x \geq 0 \)

Assume \( x^* \) has \( k \) non-zero entries, where \( k > m \). Let the indices of the non-zero entries of \( x^* \) be \( I = \{ i_1, i_2, \ldots, i_k \} \). This means that \( x^* \) can be expressed as:

\[
x^* = \begin{pmatrix}
x^*_{i_1} \\
x^*_{i_2} \\
\vdots \\
x^*_{i_k}
\end{pmatrix}
\]

The corresponding columns of \( A \) are denoted as \( A_{i_1}, A_{i_2}, \ldots, A_{i_k} \). Since \( k > m \), we have more columns than rows in the matrix formed by \( A_{i_1}, A_{i_2}, \ldots, A_{i_k} \).

By the properties of linear algebra, specifically the rank-nullity theorem, if we have more vectors (columns of \( A \)) than dimensions (rows of \( A \)), the vectors must be linearly dependent. This means there exists a non-trivial combination of these columns that sums to zero:

\[
\alpha_1 A_{i_1} + \alpha_2 A_{i_2} + \ldots + \alpha_k A_{i_k} = 0
\]

for some coefficients \( \alpha_j \) not all zero.

Now, considering the extreme point \( x^* \), it is characterized by the fact that if we take any basic feasible solution corresponding to the non-zero entries of \( x^* \), it cannot be expressed as a convex combination of other feasible solutions. The existence of a non-trivial linear combination of the columns (which are associated with the non-zero entries of \( x^* \)) that sums to zero contradicts this property of \( x^* \) being an extreme point.

Thus, if \( x^* \) has more than \( m \) non-zero entries, the columns of \( A \) corresponding to those non-zero entries must be linearly dependent, contradicting the definition of \( x^* \) being an extreme point. Therefore, we conclude that:

\[
|\{i : x^*_i > 0\}| \leq m.
\]

This shows that any extreme point \( x^* \) can have at most \( m \) non-zero entries., To solve the problem, we will formulate it as a linear program (LP) and then demonstrate that this LP can be solved in polynomial time.

### Step 1: Formulate the Linear Program

We define a variable \( p(v) \) for each vertex \( v \in V \), representing the non-negative vertex potential of vertex \( v \). The objective is to maximize the total potential across all vertices while satisfying certain constraints.

**Linear Program:**
\[
\begin{align*}
\text{max} \quad & \sum_{v \in V} p(v) \\
\text{subject to} \quad & \sum_{v \in S} p(v) \leq |E(S, \bar S)| \quad \text{for all } \emptyset \neq S \subsetneq V \\
& p(v) \geq 0 \quad \text{for all } v \in V
\end{align*}
\]

### Step 2: Show that the LP can be solved in polynomial time

We will demonstrate that the linear program described can be solved in polynomial time by using a polynomial-time separation oracle.

#### Non-negativity Constraints

The non-negativity constraints \( p(v) \geq 0 \) for all \( v \in V \) can be checked easily in \( O(|V|) \) time.

#### Other Constraints

The remaining constraints are of the form:
\[
f(S) = |E(S, \bar S)| - \sum_{v \in S} p^*(v) \geq 0 \quad \text{for all } \emptyset \neq S \subsetneq V.
\]
We need to check if there exists a violated constraint. Specifically, we need to determine if:
\[
\min_{\emptyset \subseteq S \subset V} f(S) < 0.
\]
This is equivalent to finding a cut \( S \) such that the total potential of vertices in \( S \) exceeds the number of edges crossing the cut.

#### Submodular Function Minimization

The function \( f(S) \) is a submodular function because it is the sum of the submodular cut function \( |E(S, \bar S)| \) and a linear function \( -\sum_{v \in S} p^*(v) \), which is also submodular. 

We can utilize the property of submodular functions, which states that if \( f \) is submodular, the problem of minimizing \( f \) can be solved efficiently. In our case, we need to minimize \( f(S) \) over all subsets \( S \) where \( S \) is not equal to \( V \).

To ensure we find a violated constraint, we can consider the following strategy:
1. For each vertex \( v_i \in V \), we solve the submodular minimization problem on the ground set \( V \setminus \{v_i\} \). This gives us \( n \) instances of submodular minimization.
2. Since submodular function minimization can be solved in polynomial time, we can find the minimum of \( f(S) \) for each of these instances efficiently.

### Final Algorithm

1. Formulate the LP as described.
2. Check non-negativity constraints for \( p(v) \).
3. For each vertex \( v_i \in V \), compute \( \min_{S \subset V \setminus \{v_i\}} f(S) \).
4. If any of these minimum values is less than zero, output a violated constraint. Otherwise, confirm that \( p^* \) is feasible.

### Conclusion

The algorithm runs in polynomial time because each of the \( n \) submodular minimization problems can be solved in polynomial time, and checking non-negativity constraints is also polynomial. Thus, we have a polynomial-time algorithm that finds a feasible solution \( p(v) \) satisfying the constraints of the problem while maximizing \( \sum_{v \in V} p(v) \)., source_sentence: In a nutshell, the "second album syndrome" is a theory that states that the second album of a band always sucks

You have the following regression output regarding the score_diff: the difference in scores between the second and the first album (second - first):

Dep. Variable: score_diff   
R-squared: -0.000

Interpret the 𝑅2 in your regression here. Do these analyses suggest that the "second album syndrome" exists? Why?
sentences: 1. I/O error should be a checked exception because it represents a recoverable condition related to input/output operations that applications might want to handle.
2. Backend timeout error should be a checked exception since it indicates a recoverable situation where the application could retry the operation.
3. Name already exists error should be a checked exception because it is a logical condition that applications can anticipate and manage by prompting the user for a different name., To understand why Intel Itanium contains more general-purpose registers (128) compared to the typical 32 in most RISC (Reduced Instruction Set Computing) instruction sets, we should analyze several factors step by step:

### Step 1: Purpose of Registers

Registers are small, fast storage locations within a CPU that hold data and instructions. The number of registers affects how efficiently a CPU can execute programs. More registers can reduce the need to access slower memory (like RAM), which is beneficial for performance.

### Step 2: Architectural Philosophy

Intel Itanium is based on the EPIC (Explicitly Parallel Instruction Computing) architecture, which aims to exploit instruction-level parallelism more aggressively than traditional RISC architectures. The philosophy behind EPIC encourages the processor to execute multiple instructions simultaneously, which requires additional resources, including registers.

### Step 3: Instruction-Level Parallelism

To effectively utilize instruction-level parallelism, a CPU needs to track a lot of data at once. With more registers, the Itanium architecture can keep more operands available for processing without needing to constantly load and store data from memory. This is particularly useful in complex applications where many calculations are performed in parallel.

### Step 4: Compiler Optimization

The Itanium architecture relies heavily on the compiler to optimize code for parallel execution. With more general-purpose registers available, compilers can allocate more variables and intermediate results in registers, leading to fewer memory accesses and better performance. This is contrary to RISC architectures, where the limited number of registers may force the compiler to keep more data in memory, which can slow down execution.

### Step 5: Backward Compatibility and Design Goals

Intel Itanium was designed with certain goals in mind, including high performance for server applications and complex computing tasks. The decision to implement more registers aligns with these goals, allowing it to better serve workloads that demand extensive computation and data handling.

### Step 6: Trade-offs

While having more registers can provide performance benefits, it also has trade-offs. More registers require more silicon area, which can increase cost and complexity. However, Intel Itanium was designed with the understanding that for its target market—high-performance computing—these trade-offs could be justified.

### Conclusion

In summary, the decision to include 128 general-purpose registers in Intel Itanium compared to the typical 32 in RISC designs stems from its architectural goals, focusing on maximizing instruction-level parallelism, improving compiler optimizations, and catering to high-performance computing needs. This design choice allows Itanium to handle complex workloads more efficiently, justifying the increased number of registers.

Final answer: **Intel Itanium contains more general-purpose registers (128) than most RISC instruction sets (usually 32) to maximize instruction-level parallelism, improve compiler optimizations, and cater to high-performance computing demands, allowing for more efficient execution of complex applications.**, To interpret the R-squared value in your regression output, let's break down what it means and how it relates to the concept of "second album syndrome."

1. **Understanding R-squared**: 
   - R-squared (ðÂ²) is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. An R-squared value of 1 indicates that the model explains all the variability of the response data around its mean, while an R-squared of 0 indicates that the model does not explain any of the variability.

2. **Negative R-squared**: 
   - In your output, the R-squared value is -0.000. This negative value suggests that the model is performing worse than a simple mean model (a model that predicts the mean of the dependent variable for all observations). Essentially, it means that the independent variables in your regression (if there are any) do not help to explain the differences in scores between the second and first albums.

3. **Interpreting the result**: 
   - The fact that the R-squared is essentially zero or negative indicates that there is no significant relationship found in your analysis between the scores of the first and second albums. This means that the data does not support the claim that the second album is significantly worse or better than the first album based on the scores you have.

4. **Conclusion about "second album syndrome"**: 
   - Given the R-squared value of -0.000, the analysis does not suggest that "second album syndrome" exists based on the data analyzed. If the second album consistently scored lower than the first, we would expect to see a positive R-squared value reflecting that trend. Instead, the lack of predictive power in the regression results implies that there is no discernible pattern indicating that second albums are worse than first albums in this dataset.

In summary, the regression analysis does not provide evidence to support the existence of "second album syndrome" given the negative R-squared value, indicating that the model does not explain the variance in score differences between first and second albums., source_sentence: An expression is referentially transparent if it always returns the same value, no matter
the global state of the program. A referentially transparent expression can be replaced by its value without
changing the result of the program.
Say we have a value representing a class of students and their GPAs. Given the following defintions:
1 case class Student(gpa: Double)
2
3 def count(c: List[Student], student: Student): Double =
4 c.filter(s => s == student).size
5
6 val students = List(
7 Student(1.0), Student(2.0), Student(3.0),
8 Student(4.0), Student(5.0), Student(6.0)
9 )
And the expression e:
1 count(students, Student(6.0))
sentences: To calculate the chance agreement between two annotators, you can use the formula for the chance agreement (C) based on the probability of agreement by random chance.

1. Identify the proportions of each category annotated by both annotators.
2. Use these proportions to calculate the expected agreement.

For example, if Annotator A and Annotator B both labeled a set of items into categories (e.g., positive and negative) with the following proportions:

- Annotator A: 70% positive, 30% negative
- Annotator B: 60% positive, 40% negative

The expected proportion of agreement by chance (C) can be calculated as:
\[ C = (p_A \times p_B) + (q_A \times q_B) \]
Where:
- \( p_A \) and \( q_A \) are the proportions of positive and negative labels by Annotator A
- \( p_B \) and \( q_B \) are the proportions of positive and negative labels by Annotator B

Substituting the numbers:
\[ C = (0.70 \times 0.60) + (0.30 \times 0.40) \]
\[ C = 0.42 + 0.12 = 0.54 \]

Thus, the chance agreement between the two annotators is **0.540** when rounded to three decimal places., Let's break this down simply. The function `count` takes a list of students and a specific student, then counts how many times that student appears in the list. In our example, we have a list of students with different GPAs.

When we call `count(students, Student(6.0))`, we are asking how many times a student with a GPA of 6.0 is in our list. Since we have `Student(6.0)` in the list only once, the function will return 1.

Now, to understand referential transparency: if we replace the call `count(students, Student(6.0))` with its value (which is 1), the overall result of the program would still remain the same. So, the expression is referentially transparent because it consistently gives us the same output (1) regardless of the program's state., To solve the problem of identifying a non-empty subset \( S \subsetneq V \) in a \( d \)-regular graph \( G \) using the second eigenvector \( v_2 \) of the normalized adjacency matrix \( M \), we can follow these steps:

### Step 1: Understanding Eigenvector \( v_2 \)

The second eigenvector \( v_2 \) is orthogonal to the all-ones vector \( v_1 \), indicating that it captures structural features of the graph related to its connected components. Its entries will have both positive and negative values, allowing us to partition the vertices.

### Step 2: Properties of \( v_2 \)

- The orthogonality to \( v_1 \) ensures that there are vertices with positive values (indicating one group) and negative values (indicating another group). Therefore, we can define two sets based on the sign of the entries in \( v_2 \).

### Step 3: Designing the Procedure

1. **Define the Sets:**
   - Let \( S = \{ i \in V : v_2(i) > 0 \} \).
   - Let \( T = \{ i \in V : v_2(i) < 0 \} \).

2. **Check for Non-emptiness:**
   - Since \( v_2 \) is orthogonal to \( v_1 \), at least one vertex must have a positive value and at least one must have a negative value. Hence, \( S \) cannot be empty, and \( S \neq V \).

### Step 4: Showing that \( S \) Cuts 0 Edges

We need to demonstrate that the cut defined by \( S \) does not cross any edges:

- **Edge Contributions:**
  - For any edge \( (i, j) \) in the graph, if one vertex belongs to \( S \) and the other to \( T \), the eigenvalue relationship \( M \cdot v_2 = v_2 \) indicates that the edge would create a mismatched contribution, leading to a contradiction. This implies that no edges can exist between \( S \) and \( T \).

### Final Procedure

The procedure can be summarized as follows:

```plaintext
Procedure FindDisconnectedSet(v_2):
    S = { i ∈ V : v_2(i) > 0 }
    T = { i ∈ V : v_2(i) < 0 }
    
    if S is empty:
        return T
    else:
        return S
```

### Conclusion

This algorithm ensures that we find a non-empty subset \( S \subsetneq V \) that defines a cut with no edges crossing between \( S \) and \( V \setminus S \), under the condition that \( \lambda_2 = 1 \).
pipeline_tag: sentence-similarity
library_name: sentence-transformers
metrics: cosine_accuracy@1, cosine_accuracy@3, cosine_accuracy@5, cosine_accuracy@10, cosine_precision@1, cosine_precision@3, cosine_precision@5, cosine_precision@10, cosine_recall@1, cosine_recall@3, cosine_recall@5, cosine_recall@10, cosine_ndcg@10, cosine_mrr@10, cosine_map@100
model-index: name: BGE base Financial Matryoshka
results: task: type: information-retrieval
name: Information Retrieval
dataset: name: dim 768
type: dim_768
metrics: type: cosine_accuracy@1
value: 0.2736842105263158
name: Cosine Accuracy@1
verified: no, type: cosine_accuracy@3
value: 0.7859649122807018
name: Cosine Accuracy@3
verified: no, type: cosine_accuracy@5
value: 0.8491228070175438
name: Cosine Accuracy@5
verified: no, type: cosine_accuracy@10
value: 0.9438596491228071
name: Cosine Accuracy@10
verified: no, type: cosine_precision@1
value: 0.2736842105263158
name: Cosine Precision@1
verified: no, type: cosine_precision@3
value: 0.2619883040935672
name: Cosine Precision@3
verified: no, type: cosine_precision@5
value: 0.16982456140350874
name: Cosine Precision@5
verified: no, type: cosine_precision@10
value: 0.0943859649122807
name: Cosine Precision@10
verified: no, type: cosine_recall@1
value: 0.2736842105263158
name: Cosine Recall@1
verified: no, type: cosine_recall@3
value: 0.7859649122807018
name: Cosine Recall@3
verified: no, type: cosine_recall@5
value: 0.8491228070175438
name: Cosine Recall@5
verified: no, type: cosine_recall@10
value: 0.9438596491228071
name: Cosine Recall@10
verified: no, type: cosine_ndcg@10
value: 0.6170547700367315
name: Cosine Ndcg@10
verified: no, type: cosine_mrr@10
value: 0.510181008075745
name: Cosine Mrr@10
verified: no, type: cosine_map@100
value: 0.5136000893118398
name: Cosine Map@100
verified: no, task: type: information-retrieval
name: Information Retrieval
dataset: name: dim 512
type: dim_512
metrics: type: cosine_accuracy@1
value: 0.2771929824561403
name: Cosine Accuracy@1
verified: no, type: cosine_accuracy@3
value: 0.775438596491228
name: Cosine Accuracy@3
verified: no, type: cosine_accuracy@5
value: 0.856140350877193
name: Cosine Accuracy@5
verified: no, type: cosine_accuracy@10
value: 0.9473684210526315
name: Cosine Accuracy@10
verified: no, type: cosine_precision@1
value: 0.2771929824561403
name: Cosine Precision@1
verified: no, type: cosine_precision@3
value: 0.25847953216374264
name: Cosine Precision@3
verified: no, type: cosine_precision@5
value: 0.1712280701754386
name: Cosine Precision@5
verified: no, type: cosine_precision@10
value: 0.09473684210526315
name: Cosine Precision@10
verified: no, type: cosine_recall@1
value: 0.2771929824561403
name: Cosine Recall@1
verified: no, type: cosine_recall@3
value: 0.775438596491228
name: Cosine Recall@3
verified: no, type: cosine_recall@5
value: 0.856140350877193
name: Cosine Recall@5
verified: no, type: cosine_recall@10
value: 0.9473684210526315
name: Cosine Recall@10
verified: no, type: cosine_ndcg@10
value: 0.6196869877446484
name: Cosine Ndcg@10
verified: no, type: cosine_mrr@10
value: 0.5126942355889724
name: Cosine Mrr@10
verified: no, type: cosine_map@100
value: 0.5157929794090785
name: Cosine Map@100
verified: no, task: type: information-retrieval
name: Information Retrieval
dataset: name: dim 256
type: dim_256
metrics: type: cosine_accuracy@1
value: 0.2631578947368421
name: Cosine Accuracy@1
verified: no, type: cosine_accuracy@3
value: 0.7649122807017544
name: Cosine Accuracy@3
verified: no, type: cosine_accuracy@5
value: 0.8526315789473684
name: Cosine Accuracy@5
verified: no, type: cosine_accuracy@10
value: 0.9368421052631579
name: Cosine Accuracy@10
verified: no, type: cosine_precision@1
value: 0.2631578947368421
name: Cosine Precision@1
verified: no, type: cosine_precision@3
value: 0.25497076023391807
name: Cosine Precision@3
verified: no, type: cosine_precision@5
value: 0.17052631578947366
name: Cosine Precision@5
verified: no, type: cosine_precision@10
value: 0.09368421052631577
name: Cosine Precision@10
verified: no, type: cosine_recall@1
value: 0.2631578947368421
name: Cosine Recall@1
verified: no, type: cosine_recall@3
value: 0.7649122807017544
name: Cosine Recall@3
verified: no, type: cosine_recall@5
value: 0.8526315789473684
name: Cosine Recall@5
verified: no, type: cosine_recall@10
value: 0.9368421052631579
name: Cosine Recall@10
verified: no, type: cosine_ndcg@10
value: 0.6108088502459152
name: Cosine Ndcg@10
verified: no, type: cosine_mrr@10
value: 0.5039417989417989
name: Cosine Mrr@10
verified: no, type: cosine_map@100
value: 0.5079671084661476
name: Cosine Map@100
verified: no, task: type: information-retrieval
name: Information Retrieval
dataset: name: dim 128
type: dim_128
metrics: type: cosine_accuracy@1
value: 0.2596491228070175
name: Cosine Accuracy@1
verified: no, type: cosine_accuracy@3
value: 0.7543859649122807
name: Cosine Accuracy@3
verified: no, type: cosine_accuracy@5
value: 0.8385964912280702
name: Cosine Accuracy@5
verified: no, type: cosine_accuracy@10
value: 0.9263157894736842
name: Cosine Accuracy@10
verified: no, type: cosine_precision@1
value: 0.2596491228070175
name: Cosine Precision@1
verified: no, type: cosine_precision@3
value: 0.25146198830409355
name: Cosine Precision@3
verified: no, type: cosine_precision@5
value: 0.16771929824561402
name: Cosine Precision@5
verified: no, type: cosine_precision@10
value: 0.09263157894736841
name: Cosine Precision@10
verified: no, type: cosine_recall@1
value: 0.2596491228070175
name: Cosine Recall@1
verified: no, type: cosine_recall@3
value: 0.7543859649122807
name: Cosine Recall@3
verified: no, type: cosine_recall@5
value: 0.8385964912280702
name: Cosine Recall@5
verified: no, type: cosine_recall@10
value: 0.9263157894736842
name: Cosine Recall@10
verified: no, type: cosine_ndcg@10
value: 0.6008748918945096
name: Cosine Ndcg@10
verified: no, type: cosine_mrr@10
value: 0.49439014202172044
name: Cosine Mrr@10
verified: no, type: cosine_map@100
value: 0.4987183347562729
name: Cosine Map@100
verified: no, task: type: information-retrieval
name: Information Retrieval
dataset: name: dim 64
type: dim_64
metrics: type: cosine_accuracy@1
value: 0.2596491228070175
name: Cosine Accuracy@1
verified: no, type: cosine_accuracy@3
value: 0.743859649122807
name: Cosine Accuracy@3
verified: no, type: cosine_accuracy@5
value: 0.8350877192982457
name: Cosine Accuracy@5
verified: no, type: cosine_accuracy@10
value: 0.9298245614035088
name: Cosine Accuracy@10
verified: no, type: cosine_precision@1
value: 0.2596491228070175
name: Cosine Precision@1
verified: no, type: cosine_precision@3
value: 0.24795321637426898
name: Cosine Precision@3
verified: no, type: cosine_precision@5
value: 0.1670175438596491
name: Cosine Precision@5
verified: no, type: cosine_precision@10
value: 0.09298245614035087
name: Cosine Precision@10
verified: no, type: cosine_recall@1
value: 0.2596491228070175
name: Cosine Recall@1
verified: no, type: cosine_recall@3
value: 0.743859649122807
name: Cosine Recall@3
verified: no, type: cosine_recall@5
value: 0.8350877192982457
name: Cosine Recall@5
verified: no, type: cosine_recall@10
value: 0.9298245614035088
name: Cosine Recall@10
verified: no, type: cosine_ndcg@10
value: 0.5969997633227446
name: Cosine Ndcg@10
verified: no, type: cosine_mrr@10
value: 0.48882344750765794
name: Cosine Mrr@10
verified: no, type: cosine_map@100
value: 0.4923738458546619
name: Cosine Map@100
verified: no
Likes: 0 | All-time downloads: 7
Created: 2025-06-07 17:49:25