Erovnuli Liga Qualification stats & predictions

Understanding the Football Erovnuli Liga Qualification in Georgia

The Erovnuli Liga, being the premier football league in Georgia, attracts significant attention from fans and analysts alike. The qualification rounds are particularly crucial as they determine which teams will compete in this prestigious league. Each season brings fresh matches, and with them, new opportunities for teams to showcase their skills and strategies. For enthusiasts and bettors, these matches are not just about the sport but also about predicting outcomes based on expert analysis.

The Structure of the Qualification Rounds

The qualification process for the Erovnuli Liga is meticulously organized to ensure fair competition among various clubs. Teams from lower divisions compete against each other in a series of matches that span several weeks. These rounds are designed to filter out the strongest contenders who will then advance to face off against top-tier teams in subsequent stages.

Key Factors Influencing Match Outcomes

• Team Form: The current form of a team plays a pivotal role in determining match outcomes. Teams that have been performing well in previous matches often carry momentum into their qualification games.
• Head-to-Head Records: Historical performance against specific opponents can provide insights into potential match outcomes.
• Injury Reports: The availability of key players can significantly impact a team's performance during qualification rounds.
• Tactical Approaches: Coaches often employ unique strategies tailored to exploit the weaknesses of their opponents.

Betting Predictions: An Expert's Insight

Betting on football matches requires a deep understanding of the game, including team dynamics, player conditions, and tactical nuances. Expert predictions are based on comprehensive analysis and statistical models that consider various factors influencing match outcomes.

Analyzing Team Performance

To make accurate predictions, experts analyze past performances of teams involved in the qualification rounds. This includes reviewing recent match results, goal-scoring patterns, defensive strengths, and weaknesses. Such analyses help in identifying trends that could influence future matches.

Evaluating Player Impact

Players are often the difference-makers in football matches. Experts assess individual player performances by examining statistics such as goals scored, assists provided, defensive contributions, and overall influence on the game. Key players returning from injury or suspension can also sway predictions significantly.

Tactical Considerations

The tactics employed by coaches play a crucial role in determining match outcomes. Experts evaluate coaching styles, formation changes, and strategic adjustments made during games to predict how teams might perform under different scenarios.

Daily Updates: Staying Informed with Fresh Matches

To keep up with the fast-paced nature of football qualifications, it is essential to have access to daily updates on fresh matches. These updates provide real-time information on scores, player performances, and any significant events occurring during games.

• Scores: Real-time score updates help track ongoing matches and assess their progression.
• Player Performances: Detailed reports on individual player contributions offer insights into key moments within a match.
• Injury Updates: Information about injuries sustained during games can affect team strategies and betting odds.

Leveraging Technology for Accurate Predictions

In today's digital age, technology plays an integral role in enhancing betting predictions. Advanced algorithms analyze vast amounts of data collected from past matches to forecast future outcomes with greater accuracy.

• Data Analytics: Sophisticated data analytics tools process historical data to identify patterns that might not be immediately apparent through traditional analysis methods.

Predictive Modeling Techniques

Predictive modeling involves using statistical techniques to create models that predict future events based on historical data. These models take into account various factors such as team form, player statistics, weather conditions (if applicable), and even psychological aspects like team morale or pressure situations experienced by players during crucial moments of play.

Machine Learning Algorithms

Machine learning algorithms are increasingly being used by experts for making betting predictions due to their ability to learn from large datasets without explicit programming instructions.

• Supervised Learning:This type involves training models using labeled datasets where both input features (e.g., team stats) and desired output (match outcome) are known beforehand.
  
  
• Unsupervised Learning:This approach does not require labeled data; instead,it identifies patterns within unstructured datasets like social media sentiment analysis related to teams or players.
  
  
• Natural Language Processing (NLP):NLP techniques enable computers understand human language within text sources such as news articles or social media posts regarding upcoming fixtures between competing sides.
  
  
• Sentiment Analysis:
  This technique analyzes emotions expressed through written words allowing experts gauge public opinion towards particular clubs’ chances before each fixture takes place.
  
  

Data Visualization Tools


Data visualization tools assist experts present complex datasets visually so they can easily interpret trends over time while making informed decisions when placing bets.


• Histograms:
  Histograms display frequency distribution graphs useful when analyzing variables like goals scored per match across different leagues.
  
• Pie Charts:
  Pie charts represent proportions effectively showing percentage share held by individual clubs concerning total points earned thus far within qualification rounds.
  
• Heatmaps::/Heatmaps illustrate density levels indicating areas where certain events occur frequently – e.g., location-specific scoring patterns throughout entire stadiums' playing surfaces.. The Role Of Expert Opinion In Betting Predictions
  

  Beyond technological advancements lies another critical component contributing towards successful wagering – expert opinion derived from years spent following professional sports closely combined with personal experience gained through participation at grassroots level too if applicable.

  • Judgment Calls::/Experts rely heavily upon intuition built up over time alongside analytical skills honed through continuous study which allows them make judgment calls regarding likely outcomes without solely depending upon algorithmic calculations././Understanding Contextual Nuances:/Contextual nuances refer broadly encompassing all aspects surrounding fixtures including political tensions between countries hosting international tournaments affecting fan support levels during critical knockout stages etcetera./.Fan Engagement And Its Influence On Betting Markets
    Fans play an influential role shaping betting markets through collective sentiment expressed via social media platforms where discussions about upcoming fixtures occur regularly./<8
    • /Social Media Trends:/Social media trends reflect popular opinions circulating online regarding potential victors before kickoff times thereby impacting odds offered by bookmakers./<8
    • /Fan Sentiment Analysis:/Sentiment analysis tools scan vast volumes collected text-based content generated across multiple channels extracting relevant information pertaining specifically towards public perception regarding particular clubs’ prospects./<8 The Impact Of External Factors On Match Outcomes
      A variety external factors beyond control may unexpectedly influence results such as unexpected weather conditions affecting gameplay style adopted especially outdoor venues susceptible extreme temperatures precipitation etcetera/<9
      • /Weather Conditions:/Sudden changes weather conditions like heavy rainstorms snowfall high winds etcetera can alter pitch quality drastically altering course events unfolding during live action/<9
      • /Political Situations:/Occasionally geopolitical tensions arise creating uncertainties surrounding fixtures scheduled take place affected regions potentially leading postponements cancellations altogether/<9 Ongoing Developments In Sports Analytics And Their Effect On Betting Strategies
        New developments continuously emerge sports analytics field enhancing predictive capabilities further refining existing methodologies previously utilized thereby improving accuracy rate associated prediction outputs delivered/<10
        • Integration Of Wearable Technology:: Wearable devices worn by athletes collect real-time physiological data providing valuable insights into physical condition readiness levels fatigue levels etcetera aiding analysts make more informed assessments/<10 /li/>Video Analysis Software:: Video analysis software enables detailed examination footage captured live broadcasts offering comprehensive breakdowns individual player movements tactical formations employed throughout entirety duration competitions/<10 /li/>Big Data Analytics:: Big Data analytics involves processing massive volumes structured unstructured datasets derived multiple sources enabling identification hidden patterns correlations previously unnoticed providing deeper understanding underlying dynamics shaping sporting events/<10 /li/>Artificial Intelligence (AI): AI technologies leverage machine learning algorithms neural networks natural language processing techniques enhance predictive modeling capabilities offering advanced solutions tackle complex problems encountered traditional approaches fail address adequately/<10 /li/>Blockchain Technology:: Blockchain technology holds promise revolutionizing sports betting industry ensuring transparency security integrity transactions executed decentralized ledger system eliminating fraudulent activities manipulation odds offered bookmakers/<10
          Daily Match Updates And Live Scores
          To stay updated with latest developments occurring around Football Erovnuli Liga Qualification Georgia enthusiasts rely upon reliable sources providing timely information including scores lineups substitutions tactical adjustments made throughout course proceedings live broadcast coverage available online platforms streaming services catering audience preferences varied needs interests/<11

          /li/>Official League Websites:: Official websites affiliated leagues release official statements press releases covering extensive range topics ranging fixture schedules lineup announcements injury reports disciplinary actions taken against players violating regulations rules governing competitions conducted therein/<11 /li/>Sports News Outlets:: Renowned sports news outlets publish articles blogs commentary pieces dedicated specific subjects involving Georgian football league providing insightful perspectives opinions respected journalists analysts covering respective beat areas expertise gained years following professional sport closely/<11 /li/>Social Media Platforms:: Social media platforms serve vital communication channels connecting fans worldwide facilitating exchange thoughts opinions regarding ongoing fixtures enabling real-time interaction engaging conversations taking place virtually spaces transcending geographical boundaries limitations traditionally imposed conventional methods communication historically relied upon past generations/<11 /h12>Betting Strategies For Maximizing Returns
          Betting enthusiasts employ diverse strategies aiming maximize returns minimize risks associated wagering processes involved placing bets upon uncertain outcomes inherently unpredictable nature sporting events entailed therein considering multiple factors influence eventual results observed over course timeframes specified period intervals designated predetermined timeframe duration fixed length predetermined span temporal context existence occurrence event occurrence itself subjected scrutiny evaluation assessment undertaken prior placement wagers committed financial stakes invested pursuit achieving favorable outcome desirable return investment initially sought after initiating wagering endeavor embarked undertaking commenced ventured undertaken journey begun embarked upon venture initiated pursued undertaken journey commenced embarked upon voyage embarked pursued undertaken journey commenced embarking upon voyage embarked pursuing undertaking initiated commenced embarking journey began embarkment voyage commenced embarkment pursuit initiated commencement embarkment began embarking pursuit commenced initiation embarkment voyage begun embarked pursuit initiated commencement embarkment voyage begun embarkment pursuit initiated commencement embarkment voyage begun embarked pursuit initiated commencement embarkment voyage begun embarked pursuit initiated commencement embarkment voyage begun embarked pursuit initiated commencement embarkment voyage begun embarked pursuit initiated commencement embarkment voyage begun embarked pursuit initiated commencement embarkment voyage begun embarked pursuit initiated commencement embarkment voyage begun embarked pursuit initiated commencement embarkment voyage begun embarked pursuit initiated commencement embarkment journey began embarking quest set forth underway adventure launched undertaking ventured forth quest began journey started embarking quest set forth underway adventure launched undertaking ventured forth quest began journey started embarking quest set forth underway adventure launched undertaking ventured forth quest began journey started embarking quest set forth underway adventure launched undertaking ventured forth quest began journey started embarking quest set forth underway adventure launched undertaking ventured forth quest began journey started embarking quest set forth underway adventure launched undertaking ventured forth quest began journey started embarking quest set forth underway adventure launched undertaking ventured forth endeavor set out commenced expedition launched initiative undertook endeavor undertaken mission pursued venture undertaken enterprise engaged project embraced operation embraced campaign embraced crusade embraced expedition embraced exploration embraced discovery embraced conquest embraced triumph embraced victory embraced glory embraced accomplishment achieved success attained achievement realized realization attained realization achieved realization attained realization achieved realization attained realization achieved realization attained realization achieved realization attained accomplishment accomplished achievement realized achievement realized achievement realized achievement realized achievement realized achievement realized accomplishment accomplished accomplishment accomplished accomplishment accomplished accomplishment accomplished accomplishment accomplished accomplishment accomplished accomplishment accomplished accomplishment accomplished accomplishment accomplished attainment achieved attainment achieved attainment achieved attainment achieved attainment achieved attainment achieved attainment achieved attainment achieved attainment achieved attainment acheived <|repo_name|>EthanLarson1990/chatgpt-research<|file_sep|>/prompt_response_pairs/prompt_00292.md # Prompt: Write me an article about How To Get A Job At Apple - Part Two: How To Apply For A Job At Apple . Include headings. # Response: ## How To Get A Job At Apple - Part Two: How To Apply For A Job At Apple Getting a job at Apple is highly competitive due to its reputation as one of the most innovative companies globally. However, with careful preparation and strategy execution outlined below steps increase chances securing employment position desired: ### Research Thoroughly Before applying anywhere else research extensively specific roles available within company departments aligning interests skill sets qualifications required positions applied targeting departments HR website Glassdoor forums LinkedIn profiles employees currently working there. ### Tailor Your Resume/CV Ensure resume/cv tailored job description highlighting relevant skills experience achievements accomplishments directly relating position applied emphasizing quantifiable results whenever possible demonstrating value addition potential employer organization leveraging past successes translating similar roles/projects industries environments comparable contexts situations circumstances settings surroundings environments workplaces establishments organizations corporations enterprises institutions businesses firms companies entities. ### Prepare Cover Letter Craft personalized cover letter addressing hiring manager explicitly stating reasons interest applying company showcasing enthusiasm passion commitment innovation excellence teamwork collaboration problem-solving creativity adaptability resilience perseverance determination dedication persistence perseverance ambition motivation drive initiative self-motivation self-direction self-management self-discipline self-regulation self-control autonomy independence proactivity proactiveness proactivity proactiveness proactivity proactiveness. ### Practice Interview Skills Prepare thoroughly anticipating common interview questions practicing responses articulating thoughts ideas opinions beliefs values principles convictions principles beliefs convictions principles beliefs convictions principles beliefs convictions principles beliefs clearly concisely logically coherently convincingly confidently persuasively assertively aggressively assertively aggressively assertively aggressively assertively aggressively assertively aggressively assertively aggressively assertively aggressively asserting asserting asserting asserting asserting asserting asserting asserting asserting. ### Network Effectively Utilize networking opportunities attending industry conferences meetups seminars workshops webinars symposiums summits conventions expos trade shows exhibitions fairs gatherings assemblies meetings sessions dialogues discussions conversations chats talks interviews panels debates forums roundtables focus groups brainstormings ideations think tanks think-tanks brainstorms brainstormings ideations think tanks think-tanks brainstorms brainstormings ideations think tanks think-tanks brainstorms brainstormings ideations think tanks think-tanks brainstorms brainstormings ideations think tanks think-tanks brainstorms brainstormings ideation sessions meetings engagements interactions exchanges communications correspondences correspondences correspondences correspondences correspondences correspondences correspondences correspondences correspondence correspondence correspondence correspondence correspondence correspondence correspondence correspondence. ### Follow Up Professionally After submitting application follow up professionally expressing gratitude appreciation consideration opportunity opportunity opportunity opportunity opportunity opportunity opportunity opportunity opportunity opportunity opportunity consideration consideration consideration consideration consideration consideration consideration consideration consideration consideration considered considered considered considered considered considered considered considered considered considered. ### Stay Persistent & Positive Rejection inevitable part process stay persistent positive attitude maintaining motivation determination focus energy enthusiasm passion commitment innovation excellence teamwork collaboration problem-solving creativity adaptability resilience perseverance determination dedication persistence perseverance ambition motivation drive initiative self-motivation self-direction self-management self-discipline self-regulation self-control autonomy independence proactivity proactiveness proactivity proactiveness proactivity proactiveness. By following these steps carefully increasing likelihood securing employment position desired at Apple ensuring preparedness maximizing chances success landing dream job desired career aspirations ambitions goals objectives targets aims intents purposes intents intentions intentions intentions intentions intentions intentions intentions intentions intention intent intent intent intent intent intent intent intent intent <|repo_name|>EthanLarson1990/chatgpt-research<|file_sep1]: # -*- coding: utf-8 -*- [2]: """ [3]: Created on Wed May 1 Time: am [4]: @author: Hanyu Zhang [5]: """ [6]: import os [7]: import numpy as np [8]: import pandas as pd [9]: import torch.nn.functional as F [10]: from torch.utils.data import Dataset [11]: from sklearn.model_selection import train_test_split [12]: from transformers import BertTokenizerFast [13]: import random [14]: #from bert_serving.client import BertClient [15]: #bc = BertClient() [16]: class Preprocess(object): [17]: def __init__(self): [18]: pass [19]: def preprocess(self,text): ***** Tag Data ***** ID: N/A description: This snippet represents an incomplete class definition for preprocessing, which includes initialization but lacks concrete implementation details. start line: 16 end line: 84 dependencies: - type: Class name: Preprocess start line: 16 end line:84 context description: While this snippet does not contain executable code per se, it provides a framework for preprocessing text data using libraries like `torch`, `numpy`, `sklearn`, `transformers`, etc. algorithmic depth: N/A obscurity: N/A advanced coding concepts: N/A ************ ## Challenging Aspects The provided code snippet offers several challenging aspects for students: 1. **Framework Design**: The skeleton class `Preprocess` hints at an intricate preprocessing pipeline involving various libraries (`torch`, `numpy`, `sklearn`, `transformers`). Students must design this framework comprehensively. 2. **Text Processing**: Handling text preprocessing using BERT tokenizers requires understanding tokenization intricacies—handling special tokens (`CLS`, `SEP`), padding/truncation logic. 3. **Integration**: Seamlessly integrating different libraries (like BERT tokenizers) into PyTorch workflows necessitates proficiency across these libraries. 4. **Data Handling**: Efficiently managing dataset splits while ensuring reproducibility introduces challenges around random state management. 5. **Extensibility**: Designing extensible code so additional preprocessing steps can be added easily without breaking existing functionality. 6. **Performance Optimization**: Managing memory usage efficiently when dealing with large text corpora using vectorized operations (`numpy`) or GPU acceleration (`torch`). ### Challenging Aspects in Above Code: 1. **Initialization Complexity**: Though empty now (`pass`), initializing complex objects such as BERT tokenizers should be done efficiently. 2. **Preprocessing Method**: The method signature hints at handling raw text inputs but lacks implementation details—students need to define clear preprocessing steps. 3. **Handling Dependencies**: Proper integration of dependencies like BERT tokenizer while avoiding conflicts or inefficiencies is non-trivial. 4. **Dataset Management**: Splitting datasets while preserving stratification or balancing classes adds complexity beyond simple train-test splits. ### Extension: 1. **Advanced Tokenization**: Extend tokenization logic to handle multi-language support or domain-specific vocabularies. 2. **Dynamic Dataset Loading**: Implement dynamic loading mechanisms where files can be added or removed mid-processing without halting operations. 3. **Batch Processing Enhancements**: Add batch processing capabilities with dynamic batch size adjustment based on memory constraints. 4. **Error Handling & Logging**: Introduce robust error handling mechanisms along with detailed logging for debugging complex pipelines. ## Exercise **Exercise Description**: You're tasked with expanding the [SNIPPET] provided above into a fully functional preprocessing pipeline for text classification tasks using BERT embeddings integrated within PyTorch workflows. Requirements: 1. Complete the `__init__` method to initialize necessary components like BERT tokenizer. 2. Implement the `preprocess` method: - Tokenize input text using BERT tokenizer (`BertTokenizerFast`). - Handle padding/truncation dynamically based on input lengths. - Convert tokens into tensors suitable for PyTorch model ingestion. - Return processed tensors ready for model training/inference. 3. Implement dynamic dataset loading functionality: - Continuously monitor a directory for new files being added while processing existing ones without restarting the process. - Ensure efficient memory usage when loading large datasets dynamically. 4. Implement robust error handling: - Log errors encountered during file reading/tokenization steps without terminating execution prematurely. **Note:** Assume you have access only to standard Python libraries plus those imported above (`torch`, `numpy`, `sklearn`, `transformers`). ## Solution python import os import torch.nn.functional as F import numpy as np import pandas as pd from torch.utils.data import Dataset from sklearn.model_selection import train_test_split from transformers import BertTokenizerFast class Preprocess(Dataset): def __init__(self, directory_path): super(Preprocess).__init__() self.tokenizer = BertTokenizerFast.from_pretrained('bert-base-uncased') self.directory_path = directory_path # Load initial files from directory path dynamically if needed later extensions # Monitor new files being added dynamically; use watchdog library if needed # Placeholder for storing preprocessed texts/tensors self.texts = [] self.labels = [] # Load initial files all_files = os.listdir(directory_path) for file_name in all_files: file_path = os.path.join(directory_path,file_name) if os.path.isfile(file_path): try: df = pd.read_csv(file_path) texts = df['text'].tolist() labels = df['label'].tolist() # Preprocess each row individually processed_texts = [self.preprocess(text) for text in texts] # Append processed texts/tensors along with labels self.texts.extend(processed_texts) self.labels.extend(labels) except Exception as e: print(f"Error processing file {file_name}: {str(e)}") def preprocess(self,text): """Tokenizes input text using BERT tokenizer""" try: encoded_input = self.tokenizer( text, padding='max_length', truncation=True, max_length=512, return_tensors='pt' ) return encoded_input except Exception as e: print(f"Error tokenizing text '{text}': {str(e)}") return None def __len__(self): return len(self.texts) def __getitem__(self,index): return { 'input_ids':self.texts[index]['input_ids'], 'attention_mask':self.texts[index]['attention_mask'], 'labels':self.labels[index] } # Example Usage directory_path = "path/to/dataset" preprocessor = Preprocess(directory_path) for item in preprocessor: print(item) ## Follow-up Exercise Enhance your solution further by implementing multi-threaded dynamic dataset loading using Python's threading library: 1.Implement thread-safe mechanisms ensuring no race conditions occur when accessing shared resources (like appending processed texts). 2.Expand logging functionality capturing more granular details (time-stamps) about each step's execution time. ## Solution python import threading class Preprocess(Dataset): lock = threading.Lock() def __init__(self,directory_path): ... threading.Thread(target=self.monitor_directory).start() ... def monitor_directory(self): """Monitors directory path continuously""" already_seen_files=set(os.listdir(self.directory_path)) while True: current_files=set(os.listdir(self.directory_path)) new_files=current_files-already_seen_files if new_files : already_seen_files=current_files threads=[] for file_name in new_files : thread=threading.Thread(target=self.process_file,args=(os.path.join(self.directory_path,file_name),)) thread.start() threads.append(thread) for thread in threads : thread.join() def process_file(self,file_path) : """Processes individual file""" try : df=pd.read_csv(file_path) texts=df['text'].tolist() labels=df['label'].tolist() processed_texts=[self.preprocess(text)for text in texts] with Preprocess.lock : self.texts.extend(processed_texts) self.labels.extend(labels) except Exception as e : print(f"Error processing file {file_name}: {str(e)}") This exercise ensures students deal deeply with threading issues specific only to dynamic dataset loading scenarios involving concurrent modifications while maintaining robust error handling practices. 1_79.tif) [frac{partial^{2}y}{partial x^{2}} + frac{y}{x^{2}} + frac{partial y}{partial x}left( {frac{1}{x} + k} right) + k^{2}y + lambda y^{m} = o,quad m > - l.]  *Solution* : [begin{array}{l} {y_{0}(x,k,lambda,m,l,a,c_{i})} \ {quadquadquadquadquadquadquadquadquad,,,,,,,,,,,mspace{180mu} + c_{1}exp(ikx)Phi_{m,l}(x,k,a,c_{i}) + c_{2}exp(ikx)Psi_{m,l}(x,k,a,c_{i}),} \ {Phi_{m,l}(x,k,a,c_{i}) + iPsi_{m,l}(x,k,a,c_{i}) equiv J(x;k,a,c_{i}){(kx)}^{- l},} \ {{Re}lbrack ikrbrack > l > {Re}lbrack krbrack > O,i^{2} = - l,l in C.rceil} \ \ end{array}]  ## §5° [*]. Differential equations reducible by means of two independent variable transformations. * Case I [†]. Let us suppose that equation $mathcal{E}$ admits two independent variable transformations *ξ*(*x*, *t*)*, η*(*x*, *t*) which reduce $mathcal{E}$ respectively onto equations $mathcal{E}_{I}$ , $mathcal{E}_{II}$ , i.e., let us suppose that we have two systems () [begin{matrix} {frac{partialxi}{partial x},frac{partialxi}{partial t};} & {frac{partial^{j}xi}{(partial x)^{j}},j geqslant j^{prime};} & {frac{partial^{j^{prime}}eta}{(partial t)^{j^{prime}}},j^{prime} geqslant j;} \ & {frac{partial^{k}eta}{(partial t)^{k}},k geqslant k^{prime};} & {frac{partial^{k^{prime}}xi}{(partial x)^{k^{prime}}},k^{prime} geqslant k;} \ & & \ {x,t;xi(t,x),A(t,x);eta(t,x),B(t,x);} & & \ {x,t;eta(t,x),B(t,x);xi(t,x),A(t,x);} \ {{where}mspace{180mu}} & {{wehave}mspace{180mu}} & {{used}mspace{180mu}} \ {{the}mspace{180mu}} & {{notations}mspace{180mu}} & {{of}mspace{180mu}} \ {{equations}mspace{180mu}(13)mspace{180mu}}{{and}(14).}} & {{equation}(13)mspace{-3600mu}.& {{equation}(14)mspace{-3600mu}.} end{matrix}]  ()  Let us introduce now two auxiliary functions $X(xi,t)$ , $Y(x,t)$ connected respectively according ()  to equations $mathcal{T}_{I}$ , $mathcal{T}_{II}$ . Then we shall have ()  Now let us suppose that equations $mathcal{T}_{I}$ , $mathcal{T}_{II}$ admit linearly independent solutions $X_{I},X_{II},Y_{I},Y_{II}$ . Then it follows according ()  that equation $mathcal{T}_{III}$ has linearly independent solutions $Z(x,t)$ given according ()  where ()  Now let us denote according ()  Then we shall have according () that equation $mathcal{T}_{IV}$ has linearly independent solutions $W(x,t)$ given according () where () Now let us denote according () Then we shall have according () that equation $mathcal{T}_{V}$ has linearly independent solutions $V(x,t)$ given according () Now let us suppose that equations $mathcal{T}_{I},$ , $mathcal{T}_{II}$ admit fundamental systems $X_{I},X_{II};Y_{I},Y_{II}$ . Then we shall have corresponding fundamental systems $Z,W,V$ given respectively according formulas (22)-(24). Moreover it follows accordingly formula (21) that equation $widetilde{}text{$ℰ$~}$ has linearly independent solutions given respectively according formulas (25)-(27). Thus we obtain four-dimensional space of linearly independent solutions connected respectively according formulas (28)-(30). Now let us substitute formulas (28)-(30) instead variables *u,v,w* into original differential equation ${}_{}^{}$ . Then we shall obtain differential equation connected instead variables *ξ*, *η*, *t* which admits four-dimensional space of linearly independent solutions corresponding fundamental system being connected instead variables *ξ*, *η*, *t* accordingly formulas (31)-(33). Thus we obtain reduction theorem. Theorem V (*Reduction theorem*) Let differential equation ${}_{}^{}$ admit two independent variable transformations connected instead variables (*ξ*, η)*t* accordingly formulas $(13),(14)$ . Let moreover equations ${}_{}^{},{}_{}^{}$ admit fundamental systems connected instead variables (*ξ*, η)*t* accordingly formulas $(31),(32),(33)$ . Then differential equation ${}_{}^{}$ reduces onto differential equation connected instead variables (*ξ*, η)*t*. This latter admits four-dimensional space of linearly independent solutions connected instead variables (*ξ*, η)*t* accordingly formulas $(34),(35),(36),(37)$ . For example if we put here $A=B=k,$ then we obtain formula $(34)$ reducing onto formula $(38),$ formula $(35)$ reducing onto formula $(39),$ formula $(36)$ reducing onto formula $(40),$ formula $(37)$ reducing onto formula $(41).$ Corollary V Let differential equation ${}_{}^{}$ admit two dependent variable transformations connected instead variables (*ξ*, η)*t*. Let moreover equations ${}_{}^{},{}_{}^{}$ admit fundamental systems connected instead variables (*ξ*, η)*t*. Then differential equation ${}_{}^{}$ reduces onto ordinary differential equation depending only one variable λ*(λ=t,* λ*=ξ,* λ*=η).* This latter admits four-dimensional space of linearly independent solutions depending only one variable λ. Corollary VI Let differential equation ${}_{}^{}$ admit two dependent variable transformations connected instead variables (*ξ,* η)*t*. Let moreover equations ${}_{}^{},{}_{}^{},$ admit one-dimensional spaces L*(λ=t,* λ*=ξ,* λ*=η).* Then differential equation ${}_{}^{},$ reduces onto ordinary differential equation depending only one variable λ*(λ=t,* λ*=ξ,* λ*=η).* This latter admits four-dimensional space L*(λ=t,L(λ=ξ,L(λ=η))*.* Case II Let us suppose now that transformation *(13)* transforms second-order ordinary partial derivatives ∂^{2}/∂*x* ^{−∂}* ^{*t* }into some expression containing second-order ordinary partial derivatives ∂^{∂}* ^{*f* }∂∂τ.* More precisely let us suppose that transformation *(13)* transforms second-order ordinary partial derivatives ∂^{∂}* ^{*f* }∂∂τ.* More precisely let us suppose that transformation *(13)* transforms second-order ordinary partial derivatives ∂^{∂}* ^{*f* }∂∂τ.* More precisely let us suppose that transformation *(13)* transforms second-order ordinary partial derivatives ∂^{∂}* ^{*f* }∂∂τ.* More precisely let us suppose that transformation *(13)* transforms second-order ordinary partial derivatives ∂^{∇}* ^{*f* }into some expression containing second-order ordinary partial derivatives ∇′=(*d/dτ*) (∇′=(*d/dτ*)) together with first order ordinary partial derivative d*d/dτ.* We introduce now function ψ(*f,f′*) transforming itself under substitution *(13)* into function ψ′(*g,g′*) having same form but different parameters say ψ′(*g,g′*)=*ψ(g,g′;a,b,c,d,e,f,h,k).* Moreover we assume here dependence between parameters appearing both functions ψ*(f,f′);ψ(g,g′).* Finally we assume dependence between parameters appearing both functions ψ*(f,f′);ψ(g,g′).* Finally we assume dependence between parameters appearing both functions ψ*(f,f′);ψ(g,g′).* Finally we assume dependence between parameters appearing both functions ψ*(f,f′);ψ(g,g′).* Finally we assume dependence between parameters appearing both functions ψ*(f,f ′);ψ(g,g ′).* According these assumptions transformation *(13)* connects function ψ(*f,f ′*)with function ψ’(*g,g ′*)according relation connecting parameters appearing both functions say parameter relation *(43).* Thus function ϕ(*g,g ′*) satisfies relation *(44).* Hence relation *(44)* connects coefficients appearing both sides relation *(45).* Thus relation *(45)* connects coefficients appearing both sides relation *(46).* Hence coefficient relations connect coefficients appearing both sides relation *(47).* Thus coefficient relations connect coefficients appearing both sides relation *(48).* Now putting here coefficients obtained accordingly relations connecting coefficients say obtained accordingly relations connecting coefficients say obtained accordingly relations connecting coefficients say obtained accordingly relations connecting coefficients say obtained accordingly relations connecting coefficients say obtained accordingly relations connecting coefficients say obtained accordingly relations connecting coefficients say obtained accordance coefficient relations connect coefficient connect coefficient connect coefficient connect coefficient connect coefficient connect coefficient connect coefficient connect coefficient. Theorem VI If parameter connection exists satisfying condition stated above then function φ(*g,g ′*) satisfies reduced differential equation determined uniquely once parameter connection is determined. Example I We consider here particular case where transformation reduces second-order ordinary partial derivative ∇=*d/d*x into combination containing only first order ordinary partial derivative d*d/d*t.* More precisely transformation reduces combination containing second order ordinary partial derivative d*d/d*x+d*d/d*t+dd*d*x*d*d*t+d*d*x+d*d*t+d*d*x+dd*t+dd*t+dd*t+d*x+d*t+d*x+d*t+d*x+d*t+dd*t+dd*t+dd*t.* More precisely transformation reduces combination containing combination containing combination containing combination containing combination containing combination containing combination containing combination containing combination containing combination containing d*d/d*x+d*d/d*t+dd*d*x*d*d*t+d*d*x+d*d*t+d*dx+d*dt+dxdt+xdt+xtdtxdtxdtxdtxdxdtxdtdtxdxdtxdtdtxdxdtxdtdtxdxdtxdtdtxdxdtxdtdtxdxdt+xdt+xtdtxdtxdtxdxdt+xdt+xtdtxdtxdtxdxdt+xdt+xtdtxdtxdtxdxdt+x dt+x td tx d tx dx dt xd td tx dx dt xd td tx dx dt xd td tx dx dt.* More precisely transformation reduces combination containing d*dx dd*dx dd*dx dd*dx dd*dt dd*dt dd*dt dd*dt dd*dt dd*dt dd*dt dd*dt.* More precisely transformation reduces combination containing d*dx dx d dx dt dx dt dx dt dx dt xx xx xx xx xx xx xx xx tt tt tt tt tt tt tt.* More precisely transformation reduces combination containing d*dx dx d dx dt dx dt xx xx xx tt tt.* More precisely transformation reduces combination containing d*dx dx d dx dt.* More precisely reduction theorem applies here because reduction theorem applies here because reduction theorem applies here because reduction theorem applies here because reduction theorem applies here because reduction theorem applies here because reduction theorem applies here because reduction theorem applies here because reduction theorem applies here because reduction theorem applies here because reduction theorem applies here because parameter connection exists satisfying condition stated above. Example II We consider now particular case where substitution transforms third-order mixed derivative ∇=*d*/d*x(d*/d*y(d*/dy))into third order mixed derivative *(*γ=d*/dy(d*/dy(d*/dy)). More precisely substitution transforms third order mixed derivative *(*γ=d*/dy(d*/dy(d*/dy))into third order mixed derivative *(*γ=d*/dy(d*/dy(d*/dy)). More precisely substitution transforms third order mixed derivative *(*γ=d