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In General Relativity and Quantum Gravity, diffeomorphism often called diffeomorphism invariance or general covariance is an isomorphism of differentiable manifolds that defines the physical nature of spacetime. Rather than a mathematical tool, diffeomorphism is a smooth, bijective (invertible) mapping between two open sets where both the function and its...

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𝓜𝒂𝒖𝒓𝒊𝔃𝒊𝒐 𝗜𝒃𝛼

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QUOTE 1: A worldview without feelings is a cold structure; feelings without a worldview are a blind force. : The author offers a precise and balanced insight into the relationship between intellect and emotion. A worldview without feelings becomes a cold, lifeless structure rational but empty, capable of logic yet devoid of warmth or motivation. Conversely, feelings without a coherent worldview become a blind force powerful but directionless, easily manipulated or destructive. True human maturity requires the harmonious union of both. The mind provides clarity, structure, and long-term vision. Feelings supply energy, empathy, and moral intuition. When integrated, they create a living philosophy: thought that is compassionate and emotion that is wise. This synthesis is the foundation of a complete human being. Without it, we risk becoming either heartless calculators or passionate but reckless actors. The highest expressions of humanity justice, creativity, love, and wisdom arise only when reason and feeling work together as equal partners. QUOTE 2: The unity of humanity is not a dream but a necessity: it is the only path to universal security. : The author asserts that the unity of humanity is not an idealistic dream but a fundamental necessity. In an interconnected world facing global threats climate change, pandemics, nuclear risks, resource scarcity, and technological disruption fragmented efforts and national rivalries are no longer sustainable. True universal security cannot be achieved through dominance, isolation, or temporary alliances. It requires a higher level of human solidarity: shared institutions, mutual trust, collective responsibility, and a common commitment to the survival and flourishing of our species. Without unity, every nation remains vulnerable, no matter how powerful. Unity does not mean erasing diversity or sovereignty. It means building a framework in which differences are respected while common survival imperatives are placed above them. It is the recognition that in the 21st century and beyond, humanity’s fate is collective. The path to lasting security runs through unity. Anything less is merely managed risk.

Zafar Mirzo | Quotes

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Hexagons and Octagons Those who follow football and coaching will be well aware that there are trends that emerge and become the great break through in coaching, only to vanish quite quickly. A few stick around and become a staple. Such as the rondo, or the 4v4+4 Guardiola rondo variation. One that did not stick around in the coaching collective consciousness that possible should have was Thomas Tuchel’s use of hexagonal and octagonal playing areas in training sessions. Tuchel explained that cutting off the corners and angling the pitch forced “sharp diagonal” passes that would help break the press. The positioning of players outside the hexagon/octagon or players close to the edges will be manipulated into an open body shape by the angles of the pitch. Players are impacted by environmental constraints and embodied cognition, where the geography of the playing area influences their actions. This influence spreads to the creation of triangles and diamonds within the playing area due to the “funnel” like nature of the playing area. We can use the cut outside angles by placing bounce or target players on the exterior, influencing the movement and organisation internally. The inside players will not have to move wide as those areas are occupied. The internal players will seek to create passing angles using the positioning of the outside players and their internal team mates. The diamonds and triangles will appear. If we leave the spaces on the outside empty players can move to fill the spaces. These act as free spaces to receive from the goalkeeper or open spaces for attackers to overlap into, encouraging attacking combination play and crosses. A different way of using the space is to remove goals and goalkeepers from the ends and place bounce/target players on the outside. Players now can combine with the outside players, when they do so they are then free to finish into the outside goals. The condition can be extended to combining with the target player in the opposite side of the pitch before scoring, adding an element of switching play. The hexagon and octagon are versatile spaces that help to replicate aspects of the game. By funneling the spaces we impact players body shape, ability to play forward quickly, team shape (or small group shapes), players cutting in, defending centrally, the types of combination used and the angles of line breaking pass (diagonals). The angles are hugely significant for teams that value combinations and possession football, Straight passes and receiving angles are much easier to intercept and carry high risks for being counter attacked. Short diagonals can bypass players and attacking shapes, creating angled connections. If an angles pass is given away there is still a risk of being counter attacked but there is more chance of having players around the ball to regain possession. To counter press. The question that emerges is should we then be using hexagons and octagons more? If they are of greater benefit than squares and rectangles, why use them? Should all pitches, including those of a small sided nature be hexagonal? Can the rondo square be replaced by the rondo octagon?

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