A note on two identities arising from enumeration of convex polyominoes

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On distinguishing quotients of symmetric groups

The notation used is fairly standard. We use κ, λ, µ, and ν to stand for cardinals (usually inﬁnite), and |X | for the cardinality of the set X . If Ω is any set we write Sym(Ω) for the group of all permutations of Ω (1–1 maps from Ω onto itself), with permutations acting on the right, and we write S (µ) for Sym(µ) for any cardinal µ. For g ∈ Sym(Ω) we let supp g be the support of g . If we are working in Sλ (µ)/Sκ (µ) (where Sλ (µ), Sκ (µ) are as introduced above) then we refer to sets of cardinality less than κ as small. We use overlines such as x to stand for ﬁnite sequences (‘tuples’) (x1 , x2 , . . . , xn ). By a permutation representation or action of a group G we understand a homomorphism θ from G into Sym(X ) for some set X . The representation is faithful if θ is 1–1, it is transitive if for any x, y ∈ X there is g ∈ G such that x(gθ) = y , and it is trivial if its image is the trivial group. If X is a subset (or sequence of elements) of a group G, we let X denote the subgroup generated by X . If g, h ∈ G we write g h for the conjugate h−1 gh of g by h. If g is a sequence of members of G and h ∈ G, we write g h for the h sequence whose ith entry is gi , and if g, h are sequences of members of G of the same length, we let g ∗ h be the sequence whose ith entry is gi hi . If gh 1 = g 2 for some h, g1 and g2 are said to be conjugate. If N ≤ G and f = (f1 , . . . , fn ) ∈ Gn we let N.f = (N f1 , . . . , N fn ). We write P (X ) for the power set of the set X , and Pκ (X ) for the set of subsets of X of cardinality less than κ. Then P (X ) is a boolean algebra, and each Pκ (X ) for κ inﬁnite is a ring of sets. Moreover, if ℵ0 ≤ κ < λ ≤ |X |+ , Pκ (X ) is an ideal of Pλ (X ), so we may study the quotient ring Pλ (X )/Pκ (X ), which is a boolean algebra just in the case where λ = |X |+ (that is, where Pλ (X ) = P (X )). In the remainder of this introductory section we give an outline of the main arguments of the paper. Our analysis of the quotient groups Sλ (µ)/Sκ (µ) is carried out using certain 2 many sorted structures Mκλµ and Nκλµ . (There is also a simpler version M∗ κλµ of Mκλµ applicable just in the case cf (κ) > 2ℵ0 .) These structures are devised with the object of describing the permutation action of tuples of elements of Sλ (µ), modulo small sets. The essential properties of such an n-tuple g = (g1 , g2 , . . . , gn ) are described by its action on the orbits of the subgroup g . In fact, if g1 and g 2 are n-tuples of elements of Sλ (µ) then g1 and g2 are conjugate if and only if the orbits of g 1 and g 2 can be put into 1–1 correspondence in such a way that the action of g1 on each orbit of g1 is isomorphic to that of g 2 on the corresponding orbit of g2 . Similar remarks apply in the quotient group, except that we have to allow fewer than κ ‘mistakes’ (by passing to equivalence classes of a suitable equivalence relation). These considerations lead us to observe that what should represent g in Mκλµ is a list of how many g -orbits there are of the various possible isomorphism types, where by ‘isomorphic’ here we mean ‘under the action of g’. Included among the sorts of Mκλµ are therefore, for each positive integer n, the family ISn of isomorphism types of pairs (A, f ), where f is an n-tuple of 3

关于东西方文化对我们的影响英语作文

关于东西方文化对我们的影响英语作文全文共3篇示例，供读者参考篇1The Intertwining Influences of East and WestAs a student growing up in today's increasingly globalized world, I have come to realize that the cultures of the East and West are intricately intertwined, shaping our lives in profound ways. From the foods we eat to the philosophies we embrace, the amalgamation of Eastern and Western influences has left an indelible mark on our collective identity.Let us begin by exploring the culinary realm, where the fusion of flavors and traditions is most evident. The rise of Chinese, Japanese, and Thai restaurants in Western nations is a testament to the appeal of Eastern cuisines. Conversely, the popularity of fast food chains like McDonald's and Starbucks in Asian countries reflects the Western influence on dietary habits. This cross-cultural exchange has enriched our palates, introducing us to new taste sensations and broadening our culinary horizons.Moving beyond the gastronomic sphere, the arts have been a canvas upon which the interplay of Eastern and Western cultures has been vividly depicted. The graceful movements of ballet, rooted in Western traditions, have been infused with the fluidity and elegance of Eastern dances, giving rise to innovative forms of artistic expression. Conversely, Western music has embraced the intricate rhythms and instruments of the East, creating genres like world music that transcend cultural boundaries.The realm of philosophy and spirituality has also witnessed a profound exchange between the East and West. The ancient wisdom of Eastern philosophies, such as Buddhism, Taoism, and Hinduism, has found resonance in the West, offering alternative perspectives on life, mindfulness, and inner peace. Simultaneously, Western ideologies like democracy, individualism, and human rights have influenced Eastern societies, leading to social and political transformations.In the realm of education, the influence of Eastern and Western cultures is equally palpable. While Western educational systems have traditionally emphasized critical thinking, analytical reasoning, and individual achievement, Eastern pedagogies have placed a greater emphasis on memorization, discipline, andcollective harmony. The synthesis of these approaches has given rise to innovative teaching methodologies that aim to nurture well-rounded individuals capable of navigating the complexities of a globalized world.The impact of Eastern and Western influences extends far beyond the realms mentioned above. The way we dress, the architecture we admire, and even the languages we speak bear the imprints of cultural cross-pollination. The adoption of yoga and meditation practices from the East by Westerners seeking holistic well-being is a prime example of this phenomenon. Conversely, the widespread use of Western technologies, such as smartphones and social media platforms, in Eastern societies illustrates the reciprocal nature of this cultural exchange.However, it is important to acknowledge that the intermingling of Eastern and Western cultures has not been without its challenges. Tensions arising from ideological differences, cultural misunderstandings, and power dynamics have occasionally led to conflicts and misperceptions. The West's historical colonialism and the perceived imposition of Western values on Eastern societies have been sources of contention. Likewise, the resistance to certain Western influences, such asthe preservation of traditional values and customs, has been a recurring theme in Eastern societies.Despite these challenges, the influence of Eastern and Western cultures on our lives has been overwhelmingly positive, fostering a rich tapestry of diversity and understanding. As we move forward into an increasingly interconnected future, it is imperative that we embrace the best aspects of both cultural spheres, celebrating our differences while recognizing our shared humanity.In conclusion, the intertwining of Eastern and Western cultures has profoundly shaped our identities, perspectives, and ways of life. From the foods we savor to the philosophies we contemplate, the exchange of ideas, traditions, and practices has enriched our collective experience. As students and global citizens, it is our responsibility to approach this cultural intersection with open minds and hearts, seeking to learn from one another and forge a path of mutual understanding and respect. Only through embracing the beauty and wisdom of both Eastern and Western cultures can we truly unlock our full potential as individuals and as a global community.篇2The Yin and Yang of Cultural Influence: How East and West Have Shaped Our LivesAs a student growing up in today's increasingly globalized world, I can't help but notice the profound impact that different cultures have had on shaping who we are and how we live. It's like a constant dance between the yin and yang of Eastern and Western influences, each contributing its own unique rhythms and melodies to the symphony of our modern existence.From the moment we wake up in the morning, we are bombarded with a rich tapestry of cultural elements. Many of us start our day with a practice that has its roots in ancient Eastern philosophies – meditation or yoga. These ancient traditions, born in the mystic lands of India and China, now find themselves seamlessly woven into the fabric of Western daily routines. As we quiet our minds and connect with our breath, we tap into a well of wisdom that has stood the test of millennia, reminding us of the universal human desire for inner peace and harmony.As the day progresses, we may find ourselves indulging in a cup of fragrant tea, a beverage that has been revered in Eastern cultures for centuries. The simple act of preparing and savoring this ancient elixir can transport us to the serene tea houses of Japan or the bustling markets of China, where the art of tea hasbeen perfected over generations. Yet, we often consume this Eastern delicacy while simultaneously engaging with the latest Western technological marvels – scrolling through social media on our smartphones or catching up on the news from around the globe.Our culinary experiences, too, are a delightful fusion of Eastern and Western influences. We might start our morning with a bowl of steaming oatmeal, a quintessentially Western breakfast staple, but top it with a sprinkle of chia seeds, an ancient superfood revered by the Aztecs and Mayans. For lunch, we could indulge in a delicious sushi roll, a creative amalgamation of traditional Japanese ingredients and modern Western adaptations. And for dinner, we might savor a hearty plate of pasta, a classic Italian dish, but seasoned with exotic spices like cumin and turmeric, borrowed from the vibrant culinary traditions of India.The arts and entertainment realm is another canvas upon which the brushstrokes of Eastern and Western cultures intermingle. We might spend our evenings binge-watching a gripping Korean drama, immersing ourselves in the rich storytelling traditions and intricate family dynamics that are so central to Asian narratives. Or we could lose ourselves in thelatest Hollywood blockbuster, marveling at the cutting-edge special effects and larger-than-life spectacles that have become synonymous with Western cinematic prowess.In the realm of fashion, we are truly spoiled for choice, with designers drawing inspiration from the most diverse corners of the world. One day, we might don a sleek, minimalist ensemble inspired by the clean lines and understated elegance of Japanese aesthetics. The next, we could embrace the vibrant colors and intricate embroidery of a traditional Indian sari, a garment steeped in centuries of cultural significance.Even our language reflects the interplay of Eastern and Western influences. We might pepper our conversations with the occasional "karma" or "zen," words that have their origins in ancient Sanskrit and Buddhist philosophies but have become seamlessly integrated into the English lexicon. At the same time, we might find ourselves using corporate jargon or business terminology borrowed from the West, a testament to the global reach of Western economic and cultural dominance.Of course, this cultural exchange is not a one-way street. Just as we in the West have embraced and adapted elements from the East, Eastern societies have also enthusiastically embraced aspects of Western culture. From the ubiquitous presence of fastfood chains and Western-style shopping malls in major Asian cities, to the popularity of Western music and fashion trends among the youth, the influence of the West is undeniable.This cross-pollination of cultures is not without its challenges and debates. Some argue that the homogenization of global culture is a threat to the preservation of uniquetraditions and identities. Others counter that cultural exchange is a natural and inevitable consequence of our increasingly interconnected world, and that it can enrich and expand our perspectives.Personally, I believe that the synthesis of Eastern and Western cultures is a beautiful and enriching phenomenon. It reminds us that, beneath our surface differences, we are all part of the same human tapestry, woven together by shared hopes, dreams, and experiences. By embracing the best that each culture has to offer, we can create something greater than the sum of its parts – a harmonious blend that celebrates diversity while recognizing our fundamental unity.As I look to the future, I am filled with excitement and curiosity about the ways in which our cultures will continue to intermingle and evolve. Perhaps we will see the emergence of new art forms that seamlessly blend the symbolism of Easterntraditions with the bold experimentation of Western avant-garde movements. Or maybe we will witness a renaissance in holistic wellness practices that marry the ancient wisdom of Ayurveda with cutting-edge advancements in modern medicine.Whatever lies ahead, one thing is certain: the dance between East and West will continue, each partner taking turns leading and following, dipping and twirling in a beautiful, mesmerizing choreography. And we, the lucky spectators and participants in this cultural exchange, will be enriched and transformed by the experience, our lives forever imbued with the vibrant hues and captivating rhythms of this ever-evolving global symphony.篇3The Indelible Imprint: How Eastern and Western Cultures Shape Our LivesAs students navigating the complexities of the modern world, we find ourselves at the intersection of diverse cultural influences, each leaving an indelible imprint on our worldview, values, and way of life. The interplay between Eastern and Western cultures has profoundly impacted our understanding of ourselves and our place in the global tapestry. This essay delves into the multifaceted ways in which these cultural spheres haveshaped our perspectives, highlighting their unique contributions and the synergistic potential that emerges from their harmonious coexistence.The Eastern philosophies, with their ancient roots and profound wisdom, have bequeathed to us a deep reverence for harmony, balance, and interconnectedness. The teachings of Taoism, Buddhism, and Confucianism have instilled in us the importance of living in harmony with nature, cultivating inner peace, and embracing the cyclical nature of existence. The concept of yin and yang, for instance, has taught us to appreciate the complementary duality of opposites, reminding us that light and dark, joy and sorrow, are intrinsically linked and necessary for a balanced life.Moreover, the Eastern emphasis on mindfulness and meditation has offered us invaluable tools for self-reflection and stress management in our fast-paced, modern lives. The practice of mindfulness encourages us to be fully present in the moment, cultivating a sense of clarity and equanimity amidst the constant distractions and demands of our daily routines. By embracing these Eastern philosophies, we have gained a deeper appreciation for the subtleties of human existence and the interconnectedness of all beings.On the other hand, the Western cultural traditions have endowed us with a profound respect for individual liberty, critical thinking, and scientific inquiry. The Enlightenment period, with its emphasis on reason, empiricism, and human rights, has shaped our understanding of democracy, freedom of expression, and the pursuit of knowledge. The Western philosophical traditions, from the ancient Greeks to the modern era, have encouraged us to question assumptions, engage in discourse, and continuously strive for intellectual and social progress.The Western emphasis on individualism and personal autonomy has empowered us to forge our own paths, embrace our unique identities, and assert our rights and freedoms. This cultural ethos has fostered a spirit of innovation, entrepreneurship, and artistic expression, enabling us to push boundaries and challenge the status quo. Additionally, the Western scientific tradition, with its rigorous methods and empirical approach, has equipped us with the tools to understand and harness the natural world, driving technological advancements and expanding the frontiers of human knowledge.However, it is important to acknowledge that both Eastern and Western cultures have their complexities and imperfections.The Eastern emphasis on collectivism and traditional hierarchies has, at times, stifled individual expression and perpetuated social inequalities. Conversely, the Western pursuit of individualism and material progress has occasionally come at the cost of environmental degradation and the erosion of communal bonds.As students immersed in this cultural confluence, we have the unique opportunity to selectively embrace the strengths of both traditions while critically examining their limitations. By synthesizing the Eastern principles of harmony, mindfulness, and interconnectedness with the Western ideals of critical inquiry, individual liberty, and scientific progress, we can forge a balanced and inclusive worldview that celebrates diversity while fostering unity.This cross-cultural synthesis has manifested in various domains of our lives. In education, we have witnessed the integration of mindfulness practices and holistic learning approaches alongside rigorous academic curricula and critical thinking skills. In the realm of healthcare, traditional Eastern healing modalities, such as acupuncture and herbal remedies, have found their place alongside Western allopathic medicine, offering a more comprehensive approach to well-being.Furthermore, the arts and literature have become vibrant tapestries woven from the threads of Eastern and Western storytelling traditions, philosophical perspectives, and artistic techniques. The fusion of these cultural influences has given birth to innovative artistic expressions, transcending boundaries and enriching our understanding of the human experience.As we navigate the complexities of the modern world, the influence of Eastern and Western cultures has become an integral part of our identity and worldview. We are the custodians of this rich cultural heritage, tasked with preserving the wisdom of the past while forging new paths towards a more harmonious, just, and enlightened future.In our pursuit of knowledge and personal growth, we must embrace the strengths of both traditions while remaining vigilant against their limitations. By fostering open-minded dialogue, cultural exchange, and a willingness to learn from one another, we can continue to shape a world that celebrates diversity, promotes understanding, and harnesses the collective wisdom of humanity.As students, we stand at the threshold of endless possibilities, poised to contribute our unique voices and perspectives to the ongoing tapestry of human civilization. Byembracing the invaluable lessons from Eastern and Western cultures, we can become agents of positive change, catalyzing a more harmonious, just, and enlightened world for generations to come.。

《欧洲文化入门》知识点笔记

00Greek that of the eastern half.
00230Both Latin and Greek belong to Indo-European language.
00240The Roman writer Horace(:e0lQCQMR65-8t^ WlN0) said captive Greece took her rude conqueror captive 0
00The world s first vast interior space.NLu
N,{ Colosseum('Yt珐) is an enormous.2)Yvsb_q_gRb
00410Sculpture(QX) She-wolf(
00; Father of History ! Herodotus ! war(between Greeks and Persians)
00This war is called Peleponicion wars. ZSWTY嬒d 3
00fH[ v^l g_洺0
00; The greatest historian that ever lived. ( geggO'YvS[) ! Thucydides ! war (Sparta Athens and Syracuse)
00200The burning of Corinth in 146 B.C. Marked Roman conquest of Greece.
00210The melting between Roman Culture and Greek Culture. (Wl_ g ^Jvh)
00220From 146 B.C. Latin was the language of the western half of the Roman Empire.

新视野大学英语第二册第二版第二单元单词详解

V. Detailed Study of the Text1. concernn. 1) the feeling of worry 担心，担忧concern + for/ about/ over/ withconcern + that…There is now considerable concern for their safety. 现在对他们的安全相当担心。

There is growing concern that they may have been killed. 现在越来越担心他们已遭杀害。

2) something that makes sb. worry担心的事，忧虑的事What are your main concerns as a writer? 你身为作家，对什么最感兴趣？How much money I earn is none of your concern. 我挣多少钱与你无关。

vt. 1) make sb. worried or upset 使担心、使忧虑Our losses are beginning to concern me. 我们的损失使我担起心来。

The food crisis is starting to concern African governments. 粮食危机已开始困扰非洲政府。

2) have to do with or be relevant to 涉及，与……有关系The news concerns your brother. 这消息与你兄弟有关。

He is assembling evidence concerning a murder. 他在收集一宗谋杀案的证据。

I can’t s ay the news concerns me a great deal. 我说不上这消息与我有多大关系。

[扩展] be concerned with牵涉到，与……有关，参与The letter is chiefly concerned with export commodities. 这封信主要是关于出口商品的。

OpenProblemsList

Open Problems ListArising from MathsCSP Workshop,Oxford,March2006Version0.3,April25,20061Complexity and Tractability of CSPQuestion1.0(The Dichotomy Conjecture)Let B be a relational structure.The problem of deciding whether a given relational structure has a homomorphism to B is denoted CSP(B).For which(ﬁnite)structures is CSP(B)decidable in polynomial time?Is it true that for anyﬁnite structure B the problem CSP(B)is either decidable in polynomial time or NP-complete?Communicated by:Tomas Feder&Moshe Vardi(1993) Question1.1A relational structure B is called hereditarily tractable if CSP(B )is tractable for all substructures B of B.Which structures B are hereditarily tractable?Communicated by:Pavol Hell Question1.2A weak near-unanimity term is deﬁned to be one that satisﬁes the following identities:f(x,...,x)=x and f(x,y,....y)=f(y,x,y,....y)=...=f(y,...,y,x).Is CSP(B)tractable for any(ﬁnite)structure B which is preserved by a weak near-unanimity term?Communicated by:Benoit Larose,Matt Valeriote Question1.3A constraint language1S is called globally tractable for a problem P,if P(S)is tractable,and it is called(locally)tractable if for everyﬁnite L⊆S,P(L)is tractable.These two notions of tractability do not coincide in the Abduction problem(see talk by Nadia Creignou).•For which computational problems related to the CSP do these two notions of tractability coincide?•In particular,do they coincide for the standard CSP decision problem?Communicated by:Nadia Creignou 1That is,a(possibly inﬁnite)set of relations over someﬁxed set.1Question1.4(see also Question3.5)It has been shown that when a structure B has bounded pathwidth duality the corresponding problem CSP(B)is in the complexity class NL (see talk by Victor Dalmau).Is the converse also true(modulo some natural complexity-theoretic assumptions)?Communicated by:Victor Dalmau Question1.5Is there a good(numerical)parameterization for constraint satisfaction problems that makes themﬁxed-parameter tractable?Question1.6Further develop techniques based on delta-matroids to complete the com-plexity classiﬁcation of the Boolean CSP(with constants)with at most two occurrences per variable(see talk by Tomas Feder).Communicated by:Tomas Feder Question1.7Classify the complexity of uniform Boolean CSPs(where both structure and constraint relations are speciﬁed in the input).Communicated by:Heribert Vollmer Question1.8The microstructure graph of a binary CSP has vertices for each variable/value pair,and edges that join all pairs of vertices that are compatible with the constraints.What properties of this graph are suﬃcient to ensure tractability?Are there properties that do not rely on the constraint language or the constraint graph individually?2Approximability and Soft ConstraintsQuestion2.1Is it true that Max CSP(L)is APX-complete whenever Max CSP(L)is NP-hard?Communicated by:Peter Jonsson Question2.2Prove or disprove that Max CSP(L)is in PO if the core of L is super-modular on some lattice,and otherwise this problem is APX-complete.The above has been proved for languages with domain size3,and for languages contain-ing all constants by a computer-assisted case analysis(see talk by Peter Jonsson).Develop techniques that allow one to prove such results without computer-assisted analysis.Communicated by:Peter Jonsson Question2.3For some constraint languages L,the problem Max CSP(L)is hard to approximate better than the random mindless algorithm on satisﬁable or almost satisﬁable instances.Such problems are called approximation resistant(see talk by Johan Hastad).Is a single random predicate over Boolean variables with large arity approximation resistant?What properties of predicates make a CSP approximation resistant?What transformations of predicates preserve approximation resistance?Communicated by:Johan Hastad2Question2.4Many optimisation problems involving constraints(such as Max-Sat,Max CSP,Min-Ones SAT)can be represented using soft constraints where each constraint is speciﬁed by a cost function assigning some measure of cost to each tuple of values in its scope.Are all tractable classes of soft constraints characterized by their multimorphisms?(see talk by Peter Jeavons)Communicated by:Peter Jeavons 3AlgebraQuestion3.1The Galois connection between sets of relations and sets of operations that preserve them has been used to analyse several diﬀerent computational problems such as the satisﬁability of the CSP,and counting the number of solutions.How can we characterise the computational goals for which we can use this Galois connection?Communicated by:Nadia Creignou Question3.2For any relational structure B=(B,R1,...,R k),let co-CSP(B)denote the class of structures which do not have a homomorphism to B.It has been shown that the question of whether co-CSP(B)is deﬁnable in Datalog is determined by P ol(B),the polymorphisms of the relations of B(see talk by Andrei Bulatov).Let B be a core,F the set of all idempotent polymorphisms of B and V the variety generated by the algebra(B,F).Is it true that co-CSP(B)is deﬁnable in Datalog if and only if V omits types1and2(that is,the local structure of anyﬁnite algebra in V does not contain a G-set or an aﬃne algebra)?Communicated by:Andrei Bulatov Question3.3Does every tractable clone of polynomials over a group contain a Mal’tsev operation?Communicated by:Pascal Tesson Question3.4Classify(w.r.t.tractability of corresponding CSPs)clones of polynomials of semigroups.Communicated by:Pascal Tesson Question3.5Is it true that for any structure B which is invariant under a near-unanimity operation the problem CSP(B)is in the complexity class NL?Does every such structure have bounded pathwidth duality?(see also Question1.4)Both results are known to hold for a2-element domain(Dalmau)and for majority operations(Dalmau,Krokhin).Communicated by:Victor Dalmau,Benoit Larose3Question3.6Is it decidable whether a given structure is invariant under a near-unanimity function(of some arity)?Communicated by:Benoit Larose Question3.7Let L be aﬁxedﬁnite lattice.Given an integer-valued supermodular func-tion f on L n,is there an algorithm that maximizes f in polynomial time in n if the function f is given by an oracle?The answer is yes if L is a distributive lattice(see“Supermodular Functions and the Complexity of Max-CSP”,Cohen,Cooper,Jeavons,Krokhin,Discrete Applied Mathemat-ics,2005).More generally,the answer is yes if L is obtained fromﬁnite distributive lattices via Mal’tsev products(Krokhin,Larose–see talk by Peter Jonsson).The smallest lattice for which the answer is not known is the3-diamond.Communicated by:Andrei Krokhin Question3.8Find the exact relationship between width and relational width.(It is known that one is bounded if and and only if the other is bounded.)Also,what types of width are preserved under natural algebraic constructions?Communicated by:Victor Dalmau 4LogicQuestion4.1The(basic)Propositional Circumscription problem is deﬁned as fol-lows:Input:a propositional formulaφwith atomic relations from a set S,and a clause c.Question:is c satisﬁed in every minimal model ofφ?It is conjectured(Kirousis,Kolaitis)that there is a trichotomy for this problem,that it iseither in P,coNP-complete or inΠP2,depending on the choice of S.Does this conjecturehold?Communicated by:Nadia Creignou Question4.2The Inverse Satisfiability problem is deﬁned as follows: Input:aﬁnite set of relations S and a relation R.Question:is R expressible by a CNF(S)-formula without existential variables?A dichotomy theorem was obtained by Kavvadias and Sideri for the complexity of this problem with constants.Does a dichotomy hold without the constants?Are the Schaefer cases still tractable?Communicated by:Nadia Creignou4Question4.3Let LFP denote classes of structures deﬁnable inﬁrst-order logic with a least-ﬁxed-point operator,let HOM denote classes of structures which are closed under homomorphisms,and let co-CSP denote classes of structures deﬁned by not having a homomorphism to someﬁxed target structure.•Is LFP∩HOM⊆Datalog?•Is LFP∩co-CSP⊆Datalog?(forﬁnite target structures)•Is LFP∩co-CSP⊆Datalog?(forω-categorical target structures)Communicated by:Albert Atserias,Manuel BodirskyQuestion4.4(see also Question3.2)Deﬁnability of co-CSP(B)in k-Datalog is a suﬃcient condition for tractability of CSP(B),which is sometimes referred to as having width k. There is a game-theoretic characterisation of deﬁnability in k-Datalog in terms of(∃,k)-pebble games(see talk by Phokion Kolaitis).•Is there an algorithm to decide for a given structure B whether co-CSP(B)is deﬁnable in k-Datalog(for aﬁxed k)?•Is the width hierarchy strict?The same question when B isω-categorical,but not necessarilyﬁnite?Communicated by:Phokion Kolaitis,Manuel BodirskyQuestion4.5Find a good logic to capture CSP with“nice”(e.g.,ω-categorical)inﬁnite templates.Communicated by:Iain Stewart 5Graph TheoryQuestion5.1The list homomorphism problem for a(directed)graph H is equivalent to the problem CSP(H∗)where H∗equals H together with all unary relations.•It is conjectured that the list homomorphism problem for a reﬂexive digraph is tractable if H has the X-underbar property(which is the same as having the bi-nary polymorphism min w.r.t.some total ordering on the set of vertices),and NP-complete otherwise.•It is conjectured that the list homomorphism problem for an irreﬂexive digraph is tractable if H is preserved by a majority operation,and NP-complete otherwise. Do these conjectures hold?Communicated by:Tomas Feder&Pavol Hell5Question5.2“An island of tractability?”Let A m be the class of all relational structures of the form(A,E1,...,E m)where each E i is an irreﬂexive symmetric binary relation and the relations E i together satisfy the following‘fullness’condition:any two distinct elements x,y are related in exactly one of the relations E i.Let B m be the single relational structure({1,...,m},E1,...,E m)where each E i is the symmetric binary relation containing all pairs xy except the pair ii.(Note that the relations E i are not irreﬂexive.)The problem CSP(A m,B m)is deﬁned as:Given A∈A m,is there a homomorphism from A to B m?When m=2,this problem is solvable in polynomial time-it is the recognition problem for split graphs(see“Algorithmic Graph Theory and Perfect Graphs”,M.C.Golumbic, Academic Press,New York,1980)When m>3,this problem is NP-complete(see“Full constraint satisfaction problems”,T.Feder and P.Hell,to appear in SIAM Journal on Computing).What happens when m=3?Is this an“island of tractability”?Quasi-polynomial algorithms are known for this problem(see“Full constraint satisfaction problems”,T. Feder and P.Hell,,to appear in SIAM Journal on Computing,and“Two algorithms for list matrix partitions”,T.Feder,P.Hell,D.Kral,and J.Sgall,SODA2005).Note that a similar problem for m=3was investigated in“The list partition problem for graphs”, K.Cameron,E.E.Eschen,C.T.Hoang and R.Sritharan,SODA2004.Communicated by:Tomas Feder&Pavol Hell Question5.3Finding the generalized hypertree-width,w(H)of a hypergraph H is known to be NP-complete.However it is possible to compute a hypertree-decomposition of H in polynomial time,and the hypertree-width of H is at most3w(H)+1(see talk by Georg Gottlob).Are there other decompositions giving better approximations of the generalized hypertree-width that can be found in polynomial time?Communicated by:Georg Gottlob Question5.4It is known that a CSP whose constraint hypergraph has bounded fractional hypertree width is tractable(see talk by Daniel Marx).Is there a hypergraph property more general than bounded fractional hypertree width that makes the associated CSP polynomial-time solvable?Are there classes of CSP that are tractable due to structural restrictions and have unbounded fractional hypertree width?Communicated by:Georg Gottlob,Daniel Marx Question5.5Prove that there exist two functions f1(w),f2(w)such that,for every w, there is an algorithm that constructs in time n f1(w)a fractional hypertree decomposition of width at most f2(w)for any hypergraph of fractional hypertree width at most w(See talk by Daniel Marx).Communicated by:Daniel Marx6Question5.6Turn the connection between the Robber and Army game and fractional hypertree width into an algorithm for approximating fractional hypertree width.Communicated by:Daniel Marx Question5.7Close the complexity gap between(H,C,K)-colouring and (H,C,K)-colouring (see talk by Dimitrios Thilikos)Find a tight characterization for theﬁxed-parameter tractable(H,C,K)-colouring problems.•For the(H,C,K)-colouring problems,ﬁnd nice properties for the non-parameterisedpart(H−C)that guaranteeﬁxed-parameter tractability.•Clarify the role of loops in the parameterised part C forﬁxed-parameter hardnessresults.Communicated by:Dimitrios Thilikos6Constraint Programming and ModellingQuestion6.1In a constraint programming system there is usually a search procedure that assigns values to particular variables in some order,interspersed with a constraint propagation process which modiﬁes the constraints in the light of these assignments.Is it possible to choose an ordering for the variables and values assigned which changes each problem instance as soon as possible into a new instance which is in a tractable class? Can this be done eﬃciently?Are there useful heuristics?Question6.2The time taken by a constraint programming system toﬁnd a solution toa given instance can be dramatically altered by modelling the problem diﬀerently.Can the eﬃciency of diﬀerent constraint models be objectively compared,or does it depend entirely on the solution algorithm?Question6.3For practical constraint solving it is important to eliminate symmetry,in order to avoid wasted search eﬀort.Under what conditions is it tractable to detect the symmetry in a given problem in-stance?7Notes•Representations of constraints-implicit representation-eﬀect on complexity•Unique games conjecture-structural restrictions that make it false-connectionsbetween deﬁnability and approximation•MMSNP-characterise tractable problems apart from CSP7•Migrate theoretical results to tools•What restrictions do practical problems actually satisfy?•Practical parallel algorithms-does this align with tractable classes?•Practically relevant constraint languages(”global constraints”)•For what kinds of problems do constraint algorithms/heuristics give good results?8。

陶哲轩评述佩雷尔曼的庞加莱猜想证明

2In fact, an additional argument, noting that surgery cannot destroy the simple-connectedness of the manifold components, demonstrates that the Ricci ﬂow with surgery from a simply connected manifold only involves ﬁnite unions of disconnected spheres. We thank John Morgan for clarifying this point.
1This hypothesis is in fact quite natural, because the arguments also show conversely that ﬁnite time extinction for Ricci ﬂow is only possible when the fundamental group is a free product of ﬁnite groups and inﬁnite simple groups. For the purposes of proving just the Poincar´e conjecture, though, it is possible to work entirely in the category of simply connected manifolds throughout, although this only simpliﬁes the argument at one small point (in the proof of ﬁnite time extinction). We thank John Morgan for clarifying this point.

翻译1

1.然而,如果没有阿道夫•希特勒，那就几乎可以肯定不会有第三帝国。因为阿道夫•希特勒有着恶魔般的性格、花岗岩般的意志、不可思议的本能、无情的冷酷、杰出的智力、深远的想象力以及对人和对局势的判断力。这种判断力最后由于权力和胜利冲昏了头脑而自不量力，终于弄巧成拙。
But without Adolf Hitler, who was possessed of a demoniac personality, a granite will, uncanny instincts, a cold ruthlessness, a remarkable intellect, a soaring imagination and----until toward the end, when drunk with power and success, he over reached himself----an amazing capacity to size up people and situations, there almost certainly would never have been a出生在这种地方，他有这样的母亲；这些要是让他知道了的话，他知道该多么丢人。
And he know how ashamed he would have been if he had known his mother and the kind of place in which he was born, and the kind of people among whom he was born.
7. 史密斯喝醉之后常常打老婆和女儿；第二天带着头痛发牢骚，抱怨世人不能赏识他的才华。他痛骂同行的画家都是糊涂虫。话很尖刻，有时候也有道理。

协议合同的英文缩写

协议合同的英文缩写The abbreviation of the "Agreement Contract" as a lawyer should include the following basic information: identification of the parties; their respective identities, rights, obligations, performance methods, deadlines, and default liabilities; compliance with relevant Chinese laws and regulations; clarification of the parties' rights and obligations; clarification of the legal effectiveness and enforceability, and other contents that comply with legal requirements.AC:PARTIES:Party A:Name:Address:Legalrepresentative:Nationality:Identification No.:Party B:Name:Address:Legalrepresentative:Nationality:Identification No.:IDENTITY:Party A is a [insert identity] and Party B is a [insert identity].RIGHTS AND OBLIGATIONS:1. Party A has the right to [insert rights] and is obliged to [insert obligations].2. Party B has the right to [insert rights] and isobliged to [insert obligations].3. Any other rights and obligations of the parties.PERFORMANCE METHOD AND DEADLINE:1. Party A shall perform [insert performance method] within [insert deadline].2. Party B shall perform [insert performance method] within [insert deadline].3. Any other details of performance method and deadline.DEFAULT LIABILITY:1. If Party A defaults on its obligations, it shall bear [insert default liability].2. If Party B defaults on its obligations, it shall bear [insert default liability].3. Any other details of default liability.COMPLIANCE WITH CHINESE LAWS AND REGULATIONS:The parties agree to comply with relevant Chinese laws and regulations.CLARIFICATION OF RIGHTS AND OBLIGATIONS:The parties acknowledge and agree to the clarification of their respective rights and obligations as set forth in this agreement.LEGAL EFFECTIVENESS AND ENFORCEABILITY:This agreement shall be legally effective and enforceable in accordance with the laws of the People's Republic of China.OTHERS:1. Any disputes arising from this agreement shall be settled through friendly negotiation, failing which, they shall be submitted to a competent court of law.2. Any amendments to this agreement shall be made in writing and signed by both parties.3. Any other details that comply with legal requirements.In witness whereof, the parties have executed this agreement as of the[date].Party A:Signature:Name:Date:Party B:Signature:Name:Date:。

thoughts on life and consciousness英文版

thoughts on life and consciousness英文版全文共四篇示例，供读者参考第一篇示例：Life is a mysterious journey full of surprises, challenges, and triumphs. It is a constant dance between joy and sorrow, growth and stagnation, love and loss. As we navigate through the complexities of existence, we are often left pondering the profound mysteries of life and consciousness.第二篇示例：Life is a profound mystery that has puzzled humankind for centuries. From the moment we are born until the moment we die, we are constantly searching for meaning and purpose in our existence. One of the most intriguing aspects of life is consciousness - the awareness and perception of our surroundings and ourselves. What is consciousness? How does it relate to life and our existence on this planet? These are questions that have intrigued philosophers, scientists, and spiritual leaders for millennia.第三篇示例：Life and consciousness are two intertwined concepts that have puzzled philosophers, scientists, and thinkers throughout history. The question of what defines life and how consciousness emerges is a deep and complex one that continues to be debated and explored.Life, in its most basic form, is often defined as the presence of metabolism, growth, reproduction, and response to stimuli. It is the condition that distinguishes organisms from non-living objects and is the basis for the existence of all living things on Earth. However, life is much more than just a set of biological processes - it is also a reflection of the inherent complexity and beauty of the universe.Consciousness, on the other hand, is the awareness or perception of one's own existence and the world around them. It is the ability to experience thoughts, emotions, sensations, and memories, and to have a sense of self. Consciousness is an enigmatic phenomenon that is still not fully understood by science, despite decades of research and study.The relationship between life and consciousness is a fundamental one, as consciousness is often seen as an emergent property of living systems. While it is still not clear how consciousness arises from the physical processes of the brain,many believe that it is an integral part of being alive. Without consciousness, life would lack meaning and purpose, as it is the essence of our existence and the source of our experiences and interactions with the world.One of the most profound aspects of life and consciousness is their connection to the larger universe. The fact that we are alive and conscious beings in a vast and mysterious cosmos raises questions about the nature of reality, the meaning of existence, and our place in the grand scheme of things. It is a reminder of the wonder and awe of being alive, and the responsibility that comes with it.Life and consciousness also pose ethical and philosophical challenges that we must grapple with as individuals and as a society. The understanding of what it means to be alive and conscious has profound implications for how we treat ourselves, other living beings, and the planet as a whole. It challenges us to think deeply about our values, beliefs, and actions, and to strive for a more compassionate and sustainable world.In conclusion, life and consciousness are two of the most mysterious and profound aspects of existence. They are at the core of what it means to be human, and they shape our understanding of ourselves and the world around us. They arethe source of our experiences, our thoughts, and our emotions, and they remind us of the extraordinary gift of being alive. As we continue to explore the nature of life and consciousness, let us embrace the wonder and complexity of these phenomena, and strive to live in harmony with the universe and all its inhabitants.第四篇示例：Thoughts on Life and ConsciousnessLife is a precious gift that we are given, full of wonder, beauty, and endless possibilities. Each day brings new experiences, challenges, and opportunities for growth. As we journey through life, we often find ourselves reflecting on the meaning of our existence and the nature of consciousness.Consciousness is a complex and mysterious phenomenon that has fascinated philosophers, scientists, and theologians for centuries. Our conscious experience is what gives meaning and significance to our lives, allowing us to perceive the world around us, think, feel, and act in ways that are uniquely human.One of the most profound questions that has puzzled thinkers for generations is the relationship between life and consciousness. How does consciousness arise from the physical processes of the brain? What is the nature of our subjectiveexperience of the world? These are questions that continue to challenge our understanding of the human mind.Some argue that consciousness is a purely biological phenomenon, arising from the complex workings of our brains. They believe that our thoughts, feelings, and perceptions can be explained by the firing of neurons and the release of neurotransmitters. Others propose that consciousness is something more than just our physical brains, suggesting that there may be a spiritual or metaphysical component to our existence.Regardless of the nature of consciousness, one thing is certain: it is a central aspect of our lives that shapes our identities, beliefs, and perceptions of reality. Our consciousness allows us to experience the world around us, to form relationships with others, and to make choices that shape our destinies.Life itself is a grand adventure, full of joy, sorrow, love, and loss. It is a journey that is both beautiful and challenging, filled with moments of triumph and moments of despair. It is a journey that is unique to each of us, shaped by our experiences, beliefs, and values.As we navigate the complexities of life, it is important to remember that our consciousness is a precious gift that allows usto experience the richness of the world around us. It is through our consciousness that we can connect with others, appreciate the beauty of nature, and create meaning and purpose in our lives.In the end, our thoughts on life and consciousness are deeply personal and reflective of our individual experiences and perspectives. As we continue to explore the mysteries of our existence, let us cherish the gift of consciousness and embrace the profound beauty of life in all its complexities.。

商务英语翻译英译汉第五讲省公开课一等奖全国示范课微课金奖PPT课件

5. We thank you very much for the warm hospitality accorded to our trade delegation during this visit to your country and for the close co-operation in the business discussions.
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英语中表示知觉、情欲、欲望等心理状态形容词，在系动词后做表语，往往可转译成汉语动词。常见形容词有“confident, certain, careful, cautious, angry, sure, glad, ignorant, afraid, doubtful, aware, concerned, sorry, delighted, ashamed, thankful, anxious, grateful, able…” 7) She opened the window to let fresh air in. (副词转译成汉语动词) 她把窗户打开，让新鲜空气进来。
实际上，大张旗鼓地用金钱来促使人们加入某组织或留下来，这种做法所引发激励机制上问题，远比其所处理问题要多。
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4. The main difficulty is the shortage of funds, teachers and teaching materials.
主要问题是经费不足、教员不足、教材不足这几项。
我方产品种类繁多，品质优良，价格合理，能够确保在国际市场竞争优势。（英语名词转换成汉语形容词）
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一、转译成汉语动词在英译汉时，英语中许多意思都能够用汉语
动词来表示。比如： 1）We mustn’t lose sight of the fact that air pollution is going from bad to worse in our city. 我们不能忽略这一事实，空气污染在我们城市中日益严重。 (名词转译成汉语动词) （具备动词概念名词） 2）The airline has been in operation for many years.（名词转译成汉语动词）这家航空企业已经营多年。

高一英语必修二外研版第一单元作文范文

高一英语必修二外研版第一单元作文范文全文共3篇示例，供读者参考篇1Title: Embracing Cultural Diversity: A Journey ofSelf-DiscoveryAs an inquisitive high school student, I've always been fascinated by the rich tapestry of cultures that surrounds us. Growing up in a rapidly globalizing world, I've come to realize that cultural diversity is not just a concept to be studied, but a lived experience that shapes our perspectives and enriches our understanding of the world.One of the most profound lessons I've learned is that culture is not a static entity, but rather a dynamic and ever-evolving phenomenon. It is the sum of our collective experiences, beliefs, values, and traditions that shape our identities and guide our interactions with others. When we immerse ourselves in different cultures, we embark on a journey of self-discovery, challenging our preconceived notions and broadening our horizons.My first real encounter with cultural diversity came when my family hosted an exchange student from Japan. At first, I wasstruck by the stark contrasts between our daily routines and customs. However, as time passed, I began to appreciate the beauty and depth of Japanese culture. I learned about the intricate art of calligraphy, the reverence for nature, and the emphasis on respect and harmony. Through our shared experiences, I realized that cultural differences are not barriers but opportunities to learn and grow.This realization inspired me to seek out new cultural experiences. I joined the school's international club, where I had the chance to interact with students from diverse backgrounds. We organized cultural festivals, exchanged traditional foods, and engaged in lively discussions about our respective customs and beliefs. Each encounter was a revelation, shattering stereotypes and fostering a deeper understanding and appreciation for the richness of human diversity.One particularly memorable experience was when our club organized a field trip to a local museum showcasing the art and artifacts of indigenous communities. As I gazed upon the intricate weavings, intricate pottery, and vibrant paintings, I was struck by the profound connection these artworks had to their cultural roots. I realized that art is not just a form of expression but a means of preserving and celebrating cultural heritage.Beyond the tangible aspects of culture, I've also come to appreciate the intangible elements that shape our worldviews and values. Through discussions with my classmates and teachers, I've learned about different philosophical and religious traditions, each offering unique perspectives on life's fundamental questions. These exchanges have challenged me to think critically, to question my assumptions, and to embrace the complexity of human existence.However, embracing cultural diversity is not without its challenges. I've encountered moments of misunderstanding, miscommunication, and even conflicts arising from cultural differences. Yet, these experiences have taught me the importance of open-mindedness, empathy, and respectful dialogue. By approaching cultural encounters with humility and a willingness to learn, we can bridge divides and foster mutual understanding.As I reflect on my journey thus far, I am filled with gratitude for the opportunities I've had to explore and appreciate the richness of cultural diversity. It has been a transformative experience, shaping my personal growth and deepening my appreciation for the interconnectedness of all people.Looking ahead, I am committed to continuing this journey of self-discovery and cultural exploration. I hope to study abroad, immersing myself in new cultures and gaining firsthand experiences that will further broaden my perspectives. Additionally, I aspire to use my understanding of cultural diversity to promote inclusivity and bridge divides within my community, fostering an environment where all cultures are celebrated and respected.In a world that is increasingly interconnected, embracing cultural diversity is not just a choice but a necessity. It is through understanding and appreciating our differences that we can build a more compassionate, just, and harmonious society. As a student and a global citizen, I am grateful for the opportunities to learn, grow, and contribute to this ongoing journey of cultural exchange and understanding.篇2Unit 1: Life and HealthEssay Title: The Importance of Maintaining a Balanced LifestyleAs a high school student, I often find myself overwhelmed with the pressures of academics, extracurricular activities, andsocial life. It's easy to get caught up in the whirlwind of responsibilities and lose sight of what truly matters – our physical and mental well-being. However, through personal experience and observation, I've come to realize the immense importance of maintaining a balanced lifestyle.In today's fast-paced world, many of us have embraced a sedentary lifestyle, spending countless hours glued to our screens or buried in textbooks. While academic achievement is undoubtedly crucial, neglecting our physical health can have detrimental consequences. Regular exercise not only keeps our bodies fit but also boosts our mental alertness and focus, enabling us to perform better academically.I vividly remember the times when I would spend hours upon hours studying, fueled by caffeine and unhealthy snacks, only to feel drained and sluggish the next day. It wasn't until I incorporated regular exercise into my routine that I truly understood the profound impact it had on my overall well-being. Whether it's a brisk walk, a game of basketball with friends, or a session at the gym, engaging in physical activity has become an essential part of my day.Moreover, maintaining a balanced diet is equally important. As students, it's tempting to rely on fast food and energy drinksto power through long study sessions, but these choices can have adverse effects on our health in the long run. I've learned to prioritize wholesome, nutrient-dense meals that provide sustained energy and nourishment for my growing body and mind.Beyond physical health, a balanced lifestyle also encompasses mental and emotional well-being. The constant pressure to excel academically and meet societal expectations can take a toll on our mental health. It's crucial to find healthy outlets for stress relief, whether it's through hobbies, meditation, or simply spending quality time with loved ones.One of the most valuable lessons I've learned is the importance of setting boundaries and practicing self-care. It's okay to say no to commitments that stretch us too thin and to carve out time for activities that bring us joy and relaxation. By nurturing our mental and emotional well-being, we can cultivate resilience and better cope with the challenges that life throws our way.Furthermore, a balanced lifestyle also involves fostering meaningful relationships and maintaining a strong support system. As social beings, we thrive on connections with others. Surrounding ourselves with positive, uplifting individuals whoshare our values and goals can provide a sense of belonging and motivation.Looking back on my high school journey so far, I've come to appreciate the importance of striking a balance between academic pursuits, physical health, mental well-being, and social connections. It's a delicate equilibrium that requires conscious effort and discipline, but the rewards are invaluable.As I look towards the future, I am determined to carry these lessons with me into adulthood. A balanced lifestyle is not just a temporary pursuit but a lifelong commitment to personal growth, fulfillment, and overall well-being. By prioritizing our physical, mental, and emotional needs, we can unlock our full potential and navigate the complexities of life with grace and resilience.In conclusion, the path to success and happiness lies not in sacrificing one aspect of our lives for another but in finding the perfect harmony among them all. It's a journey of self-discovery, discipline, and unwavering commitment – a journey that I wholeheartedly embrace as a high school student and beyond.篇3Title: The Transformative Power of ReadingReading has been an integral part of my life since I was a young child. From the vibrant picture books my parents would read to me at bedtime to the captivating novels that ignited my imagination, the written word has always held a special place in my heart. As I embarked on my journey through high school, the significance of reading has only grown more profound, shaping not only my academic pursuits but also my personal growth and understanding of the world around me.One of the most remarkable aspects of reading is its ability to transport us to worlds beyond our immediate reality. Through the pages of a book, we can traverse vast landscapes, immerse ourselves in foreign cultures, and even travel through time itself. This escapism, far from being a mere form of entertainment, has proven to be a powerful tool for expanding our horizons and challenging our preconceived notions.I vividly recall the first time I delved into the realms of classical literature, encountering the timeless works of Shakespeare, Austen, and Dickens. Initially, the archaic language and complex themes posed a formidable challenge, but as I persevered, a new world unfolded before me. I found myself drawn into the intricate tapestry of human emotions, societal norms, and the enduring themes that transcend time and place.Through the eyes of these literary giants, I gained invaluable insights into the human condition, cultivating a deeper sense of empathy and understanding for the multifaceted experiences that shape our existence.Moreover, reading has played a pivotal role in honing my critical thinking skills – a invaluable asset in the ever-evolving landscape of academia and beyond. As I navigated through various genres and styles, I learned to analyze complex narratives, dissect the author's intent, and formulate my own interpretations. This process of active engagement with the text has not only enriched my analytical abilities but has also taught me the importance of considering multiple perspectives and questioning assumptions.Furthermore, reading has proven to be an invaluable resource in broadening my knowledge across a multitude of disciplines. From delving into the intricacies of scientific discoveries to exploring the rich tapestry of historical events, books have served as a gateway to a world of information and ideas. This exposure has not only deepened my understanding of various subjects but has also fostered a sense of intellectual curiosity that continues to drive my thirst for learning.Beyond its academic merits, reading has also profoundly impacted my personal growth and development. Through the compelling narratives and relatable characters, I have gained insights into the complexities of human nature, confronted moral dilemmas, and grappled with existential questions. These literary explorations have challenged me to reflect on my own values, beliefs, and aspirations, ultimately shaping the person I strive to become.Additionally, reading has cultivated my appreciation for the power of language and the art of storytelling. By immersing myself in the intricate tapestry of words woven by skilled authors, I have developed a deeper respect for the craft of writing and the ability to convey emotions, ideas, and experiences through the written word. This newfound appreciation has not only enhanced my enjoyment of literature but has also inspired me to explore my own creative inclinations, whether through poetry, short stories, or personal essays.As I look back on my high school journey, I am filled with gratitude for the transformative power of reading. It has been a constant companion, a source of inspiration, and a gateway to knowledge and personal growth. However, I recognize that myjourney with the written word is far from over. As I embark on the next chapter of my life, I carry with me the lessons, insights, and。

文化英文作文范文

文化英文作文范文Title: Exploring Cultural Diversity: A Journey of Understanding。

In our rapidly globalizing world, the significance of understanding and appreciating cultural diversity cannot be overstated. Cultures shape our identities, beliefs, and behaviors, yet often, we overlook the richness they bringto our lives. Through this essay, let us delve into the depths of cultural diversity and explore its profoundimpact on individuals and societies.Firstly, embracing cultural diversity fosters mutual respect and empathy among people from different backgrounds. When we take the time to learn about others' customs, traditions, and values, we gain insight into their perspectives and experiences. This understanding lays the foundation for meaningful connections and collaborations, transcending boundaries of nationality, ethnicity, and religion.Moreover, cultural diversity stimulates creativity and innovation. Exposing ourselves to diverse ways of thinking and problem-solving expands our cognitive horizons, enabling us to approach challenges from multiple angles. In a globalized economy driven by creativity and innovation, organizations that embrace diversity are better equipped to adapt to change and thrive in an ever-evolving landscape.Furthermore, cultural diversity enriches our social fabric, infusing it with a tapestry of languages, cuisines, arts, and customs. Each culture contributes unique flavors, melodies, and colors to the collective human experience, creating a vibrant mosaic of diversity. Celebrating this diversity not only preserves cultural heritage but also cultivates a sense of belonging and inclusivity for all members of society.However, despite its numerous benefits, cultural diversity also presents challenges, particularly in the form of cultural misunderstandings and conflicts. Misinterpretations arising from differences incommunication styles, social norms, and values can lead to tension and discord between individuals and communities. Therefore, promoting intercultural dialogue and fostering cultural competence are essential for navigating these challenges and building harmonious relationships across cultures.In today's interconnected world, technology plays a pivotal role in facilitating cross-cultural exchange and understanding. Social media platforms, online forums, and virtual reality simulations offer opportunities for people to connect with individuals from diverse cultural backgrounds, transcending geographical barriers and fostering empathy and mutual respect.Educational institutions also play a crucial role in promoting cultural diversity and fostering intercultural competence among students. By integrating multicultural curricula, language immersion programs, and exchange initiatives into their educational offerings, schools can equip students with the knowledge, skills, and attitudes needed to thrive in a multicultural world.In conclusion, cultural diversity is not merely a source of difference; it is a source of strength, resilience, and enrichment for individuals and societies alike. By embracing diversity, fostering intercultural dialogue, and promoting cultural competence, we can build a more inclusive and equitable world where every voice is heard and valued. As we embark on this journey of understanding, let us celebrate the beauty of our differences and strive to create a future where diversity is not only accepted but embraced as a fundamental aspect of our humanity.。

大学情侣英语作文

大学情侣英语作文College Couple。

College life is a time of exploration, growth, and new experiences. For many students, it is also a time to find love and companionship. College couples are a common sight on campuses around the world, and their relationships often play a significant role in their overall college experience.In my opinion, being part of a college couple can be both rewarding and challenging. On the one hand, having a partner to share your college journey with can provide emotional support, companionship, and a sense of belonging. It can also be a source of motivation and inspiration, as you both strive to achieve your academic and personal goals. Additionally, being in a relationship during college can help you develop important life skills such as communication, compromise, and time management.On the other hand, being in a college couple can alsopresent its fair share of challenges. Balancing a relationship with the demands of college life, such as classes, assignments, and extracurricular activities, can be difficult at times. Additionally, college couples may face external pressures from friends, family, and society, which can sometimes put a strain on their relationship. It is important for college couples to be mindful of these challenges and to work together to overcome them.I believe that communication is key in any college relationship. It is important for both partners to be open and honest with each other about their needs, expectations, and concerns. This can help prevent misunderstandings and conflicts from arising, and can also strengthen the bond between the couple. Additionally, finding a balance between spending quality time together and maintaining individual identities is crucial for a healthy college relationship.In my own experience, being part of a college couple has been a positive and enriching experience. My partner and I have supported each other through the ups and downs of college life, and have created many cherished memoriestogether. We have learned to communicate effectively, compromise when necessary, and appreciate each other's strengths and weaknesses. Our relationship has added depth and meaning to our college experience, and has helped us grow as individuals.In conclusion, being part of a college couple can be a fulfilling and transformative experience. It can provide emotional support, companionship, and personal growth, while also presenting its fair share of challenges. By communicating openly, finding a balance, and working together, college couples can navigate the complexities of their relationships and create a strong foundation for their future together.。

异族通婚看法英语作文

异族通婚看法英语作文Title: Intermarriage: A Cross-Cultural Perspective。

In recent years, intermarriage, or marriage between individuals from different cultural or ethnic backgrounds, has become increasingly common. This phenomenon has sparked various debates and discussions globally. While some advocate for the benefits of intercultural unions, others express concerns about potential challenges and consequences. In this essay, we will explore different perspectives on intermarriage and analyze its implications.To begin with, proponents of intermarriage often emphasize its role in promoting cultural exchange and understanding. Marrying someone from a different cultural background can facilitate exposure to new customs, traditions, and perspectives. This exposure fosters tolerance, empathy, and appreciation for diversity, contributing to a more inclusive and harmonious society. Additionally, intercultural couples serve as livingexamples of love transcending cultural boundaries, challenging stereotypes, and promoting social cohesion.Moreover, intermarriage can enrich family dynamics and personal growth. Children born into intercultural families often have the opportunity to embrace and celebratemultiple cultural identities. Growing up in such environments can enhance their adaptability, open-mindedness, and language skills. Furthermore, navigating the complexities of two cultures can instill resilience and a broader worldview, equipping individuals with valuable life skills.However, critics of intermarriage raise valid concerns regarding its potential challenges. One common concern is the preservation of cultural heritage and identity. Intermarriage may lead to the dilution or loss of cultural traditions, languages, and practices within families. This loss can evoke feelings of disconnection or alienation among future generations, especially if efforts to transmit cultural heritage are insufficient. Additionally, clashes arising from cultural differences, such as communicationstyles or value systems, can strain marital relationships and family cohesion.Furthermore, intermarriage may exacerbate existingpower dynamics and inequalities between cultures. In some cases, individuals from marginalized or underrepresented communities may face pressures to assimilate into the dominant culture, eroding their sense of identity and autonomy. Moreover, disparities in social status, economic resources, or legal rights between partners from different backgrounds can create imbalances within relationships, leading to issues of exploitation or discrimination.Despite these challenges, it is essential to recognize that intermarriage is a complex and multifaceted phenomenon, influenced by various factors such as historical context, socioeconomic status, and individual motivations. Rather than adopting a one-size-fits-all approach, it is crucialto consider the unique circumstances and experiences ofeach intercultural couple.In conclusion, intermarriage reflects theinterconnectedness of our increasingly globalized world, presenting both opportunities and challenges. While it can promote cultural exchange, social integration, and personal growth, it also raises concerns about identity preservation, power dynamics, and inequality. Ultimately, fosteringmutual respect, understanding, and equitable partnershipsis essential for navigating the complexities ofintercultural relationships and building a more inclusive society.。

夫妻之间英文作文

夫妻之间英文作文Title: The Dynamics of Marriage: Navigating Love, Challenges, and Growth。

Marriage, an institution that binds two individuals in a bond of love, commitment, and partnership, is often described as a journey filled with highs and lows, joys and challenges. It is a sacred union that requires dedication, understanding, and compromise from both partners. In this essay, we delve into the intricacies of marriage, exploring its essence, the dynamics between spouses, and the keys to a successful and fulfilling marital relationship.At the heart of every marriage lies love – a deep, profound, and enduring love that forms the foundation of the relationship. Love is what brings two people together, igniting the spark that blossoms into a lifelong companionship. It is a force that transcends boundaries, sustains through adversity, and grows stronger with each passing day. In a marriage, love serves as the guidinglight, illuminating the path through the darkest of times and celebrating the brightest of moments.However, love alone is not enough to sustain a marriage. Alongside love, there must exist mutual respect, trust, and communication – the cornerstones of a healthy and thriving relationship. Respect entails valuing each other's opinions, beliefs, and differences, recognizing the inherent worthand dignity of your partner. Trust is the bedrock uponwhich a marriage is built, fostering a sense of security, reliability, and fidelity. Communication, open and honest, serves as the lifeline of the relationship, enablingcouples to express their thoughts, feelings, and concerns freely.Yet, despite the presence of love, respect, trust, and communication, challenges are inevitable in any marriage. From financial struggles to differences in values, from conflicts in parenting styles to external pressures,couples must navigate a myriad of obstacles together. It is during these challenging times that the true strength ofthe marital bond is tested – requiring patience,understanding, and compromise from both partners. Rather than viewing challenges as threats to the relationship, couples should see them as opportunities for growth, learning, and deeper connection.One of the keys to a successful marriage is the ability to navigate conflicts and disagreements effectively. Conflict is a natural and inevitable part of any relationship, arising from differences in personalities, expectations, and experiences. However, it is not the presence of conflict that determines the health of a marriage, but rather how couples manage and resolve it. Effective conflict resolution involves active listening, empathy, and a willingness to find common ground. It requires setting aside ego and defensiveness, prioritizing the well-being of the relationship above individual needs and desires.Moreover, maintaining a sense of individuality within the context of marriage is essential for personalfulfillment and relationship satisfaction. While marriage signifies the merging of two lives into one, it isimportant for each partner to retain a sense of self –pursuing personal interests, goals, and aspirations. By nurturing their own identities and passions, spouses can bring richness, diversity, and vitality to the relationship, fostering a deeper sense of connection and appreciation for each other.In conclusion, marriage is a profound andtransformative journey that requires love, dedication, and mutual effort from both partners. It is a journey marked by moments of joy and sorrow, triumphs and challenges, growth and evolution. By cultivating love, respect, trust, and communication, navigating conflicts effectively, and maintaining a sense of individuality, couples can forge a strong and enduring marital bond that stands the test of time. As they embark on this journey together, hand in hand, they embark on a voyage of discovery, companionship, andlove that will enrich and fulfill their lives in ways beyond measure.。

对赌协议的英文

对赌协议的英文Gambling AgreementThis Agreement is made on [date] between [Party A] and [Party B] (collectively, “Parties”).1. Basic InformationParty A:[Name][Address][Representative][Identification Number]Party B:[Name][Address][Representative][Identification Number]2. Parties’ Identities, Rights, Obligations, Performance Methods, Terms, and Breach Liability2.1 Party A and Party B agree to gamble with each other on [event or activity] (the “Gambling Activity”).2.2 Party A and Party B agree to pay [amount] to each other if they lose the Gambling Activity.2.3 Party A and Party B agree to execute this Agreement in accordance with Chinese laws and regulations.2.4 Party A and Party B agree to comply with the relevant laws and regulations in China related to this Gambling Agreement.2.5 Each Party shall have the right to request the other Party to perform its obligations under this Agreement.2.6 In case of dispute arising from this Agreement, the Parties agree to submit the dispute to [arbitration or court] with jurisdiction in [city/province/country].2.7 The liability for breach of this Agreement shall be borne by the Party who breaches it.3. Rights and Obligations3.1 Party A and Party B shall perform their respective obligations in good faith.3.2 Party A and Party B shall keep the details of this Agreement confidential.3.3 Party A and Party B shall not disclose any confidential information of the other Party to any third party without the other Party’s written consent.3.4 Party A and Party B shall not disclose any confidential information obtained from the Gambling Activity to any third party without the other Party’s written consent.3.5 Party A and Party B shall not disclose any confidential information related to the Gambling Activity to any third party without the other Party’s written consent.4. Legal Effectiveness and Enforcement4.1 This Agreement shall be legally binding on the Parties.4.2 This Agreement shall be enforceable in accordance with the laws and regulations of China.4.3 The Parties acknowledge that the terms of this Agreement are valid, legal, and enforceable.4.4 The Parties acknowledge that they have the legal capacity to enter into this Agreement.4.5 The Parties shall cooperate with each other to execute this Agreement as soon as possible.4.6 The Parties shall not assign or transfer their rights and obligations under this Agreement to any third party without the other Party's written consent.4.7 Any amendment, supplement, modification, or termination of this Agreement shall be in writing and signed by both Parties.5. Miscellaneous5.1 This Agreement shall take effect from the date of signing by both Parties.5.2 This Agreement is executed in Chinese and English languages. In the event of any conflict between the two languages, the Chinese version shall prevail.5.3 This Agreement represents the entire understanding between the Parties and supersedes all prior negotiations, understandings, and agreements between the Parties.5.4 This Agreement may be executed in counterparts, each of which shall be deemed an original, but all of which when taken together shall constitute one and the same instrument.5.5 This Agreement shall be governed by and construed in accordance with the laws of China.In witness whereof, the Parties have executed this Agreement on the date first above written.[Party A]Signature:Name:Title:[Party B]Signature:Name:Title:。

推进全球化发展英语作文

Globalization has become a buzzword in the 21st century, often discussed in classrooms, boardrooms, and political arenas. As a high school student who has grown up in an era where the world is more interconnected than ever before, I have witnessed firsthand the profound impact of globalization on various aspects of life. In this essay, I will share my perspective on how globalization has been advancing and the benefits it brings to our world.Growing up in a small town, I was initially shielded from the direct effects of globalization. However, as I delved deeper into my studies and became more aware of the world outside my immediate surroundings, I began to see the threads of globalization woven into the fabric of our daily lives. One of the most evident aspects of globalization is the ease with which we can now access goods and services from around the world. A simple trip to the local supermarket reveals an array of products from different countries, from Italian pasta to Thai coconut milk, showcasing the diversity and availability brought by global trade.The advancements in technology have been a driving force behind the rapid pace of globalization. The internet, in particular, has revolutionized the way we communicate, learn, and do business. As a student, I have benefited immensely from online resources that have broadened my educational horizons. I can access lectures from renowned professors across the globe, participate in international student forums, and collaborate on projects with peers from different cultures. This interconnectedness has not only enriched my learning experience but also instilled in me a sense of global citizenship.Moreover, globalization has facilitated cultural exchange and understanding. Through social media, movies, music, and literature, we are exposed to diverse cultures and perspectives that challenge our preconceived notions and promote tolerance. I remember the excitement of watching a foreign film festival at my school, where we were introduced to the rich narratives and artistic expressions of various countries. These experiences have helped me appreciate the beauty of cultural diversity and the importance of preserving our unique cultural identities while embracing global influences.Economically, globalization has opened up new markets and opportunities for businesses, leading to increased competition and innovation. Companies can now reach customers in different parts of the world, and consumers have access to a wider range of products at competitive prices. This has been particularly beneficial for developing countries, which have been able to tap into global markets, attract foreign investments, and improve their economic standing.However, it is important to acknowledge that globalization also presents challenges. Issues such as job displacement due to offshoring, environmental concerns arising from increased production and transportation, and the potential loss of cultural identity are valid concerns that need to be addressed. As a high school student, I am aware of these challenges and believe that it is our collective responsibility to find solutions that promote sustainable and equitable globalization.In conclusion, globalization is a multifaceted phenomenon that has significantly impacted our world. It has brought about numerous benefits, such as increased access to goods and services, enhanced cultural exchange, and economic growth. However, it also presents challenges that require thoughtful and collaborative solutions. As the next generation, it is our duty to harness the positive aspects of globalization while addressing its shortcomings to create a more interconnected and prosperous world.。

新视野大学英语第二版第二册课文翻译Unit3-SectionB

Unit 3Secti‎o n BRich Meeti‎n g His Futur‎e Mothe‎r-in-lawAfter‎much thoug‎h t, I came up with a brill‎i ant plan for Rich to meet my mothe‎r and win her over. In fact, I arran‎g ed it so my mothe‎r would‎want to cook a meal espec‎i ally‎for him.One day, my mothe‎r calle‎d me, to invit‎e me to a birth‎d ay dinne‎r for my fathe‎r. My broth‎e r Vince‎n t was bring‎i ng his girlf‎r iend‎, Lisa Lum. I could‎bring‎a frien‎d, too.I knew she would‎do this, becau‎s e cooki‎n g was how my mothe‎r expre‎s sed her love, her pride‎, her power‎,her proof‎that she knew more than any one else. "Just be sure to tell her later‎that her cooki‎n g was the best you ever taste‎d," I told Rich. "Belie‎v e me."The eve of the dinne‎r, I sat in the kitch‎e n watch‎i ng her cook, waiti‎n g for the right‎momen‎t to tell her about‎our marri‎a ge plans‎, that we had decid‎e d to get marri‎e d next July, about‎seven‎month‎s away. She was cubin‎g garli‎c and slici‎n g cabba‎g e into small‎piece‎s and chatt‎i ng at the same time about‎Aunti‎e Suyua‎n: "She can only cook looki‎n g at direc‎t ions‎. My instr‎u ctio‎n s are in my finge‎r s.I know what secre‎t ingre‎d ient‎s to put in just by using‎my nose!" And she was slici‎n g so quick‎l y, seemi‎n gly not payin‎g atten‎t ion to her sharp‎chopp‎i ng knife‎, that I was afrai‎d the tips of her finge‎r s would‎becom‎e one of the ingre‎d ient‎s of the purpl‎e veget‎a ble and pork dish.I was hopin‎g she would‎say somet‎h ing first‎conce‎r ning‎Rich. I had seen her expre‎s sion‎when she opene‎d the door, her force‎d smile‎as she surve‎y ed him from head to toe, check‎i ng her judgm‎e nt of him again‎s t that alrea‎d y given‎to her by Aunti‎e Suyua‎n. I tried‎to antic‎i pate‎what criti‎c isms‎she would‎have.Rich was not only not Chine‎s e, he was also my junio‎r, a few years‎young‎e r than I was. And unfor‎t unat‎e ly, he looke‎d much young‎e r with his curly‎red hair, smoot‎h pale skin, and the splas‎h of orang‎e freck‎l es acros‎s his nose. He was a bit on the short‎side, compa‎c tly built‎.In his dark busin‎e ss suits‎,he looke‎d nice but easil‎y forge‎t tabl‎e, like someb‎o dy's nephe‎w at a funer‎a l. This was why I didn't notic‎e him the first‎year we worke‎d toget‎h er at the firm. But, my mothe‎r notic‎e d every‎t hing‎."So what do you think‎of Rich?" I final‎l y asked‎, holdi‎n g my breat‎h.She tosse‎d the garli‎c in the hot oil which‎bubbl‎e d in a loud, angry‎sound‎. "So many spots‎on his face," she said.I could‎feel the goose‎bumps‎rise on my back. "They're freck‎l es. Freck‎l es are good luck, you know," I felt compe‎l led to defen‎d on his behal‎f, a bit too heate‎d ly as I raise‎d my voice‎above‎the noise‎of the kitch‎e n."Oh?" she said innoc‎e ntly‎."Yes, the more spots‎the bette‎r. Every‎b ody knows‎that."She consi‎d ered‎this a momen‎t and then smile‎d and spoke‎in a Chine‎s e diale‎c t: "Maybe‎this is true. When you were young‎,you got the chick‎e n pox. So many spots‎,you had to stay home for ten days. So lucky‎, you thoug‎h t."I could‎n't save Rich in the kitch‎e n. And I could‎n't save him later‎at the dinne‎r table‎eithe‎r.He had broug‎h t a bottl‎e of Frenc‎h wine, somet‎h ing he did not know my paren‎t s could‎not appre‎c iate‎. My paren‎t s did not even own appro‎p riat‎e glass‎e s for wine. And then he also made themista‎k e of drink‎i ng not one but two frost‎e d glass‎e s full, while‎every‎b ody else had a half-inch "just for taste‎."But the worst‎happe‎n ed when Rich criti‎c ized‎my mothe‎r's cooki‎n g, and he didn't even have a clue about‎what he had done. As is the Chine‎s e cook's custo‎m, my mothe‎r alway‎s made negat‎i ve remar‎k s about‎her own cooki‎n g. That night‎she chose‎to direc‎t it towar‎d her famou‎s steam‎e d pork and prese‎r ved veget‎a ble dish, which‎she alway‎s serve‎d with speci‎a l pride‎."Ai! This dish not salty‎enoug‎h, no flavo‎r," she compl‎a ined‎, after‎tasti‎n g a small‎bite. "It is too bad to eat."This was our famil‎y's cue to eat some and procl‎a im it the best she had ever made. But befor‎e we could‎be so diplo‎m atic‎, Rich said, "Y ou know, all it needs‎is a littl‎e soy sauce‎." And he proce‎e ded to pour a river‎f ul of the salty‎black‎stuff‎on the china‎plate‎, right‎befor‎e my mothe‎r's shock‎e d eyes. And even thoug‎h I was hopef‎u l throu‎g hout‎the dinne‎r that my mothe‎r would‎someh‎o w see Rich's kindn‎e ss, his sense‎of humor‎and charm‎, I knew he had faile‎d miser‎a bly in her eyes.Rich obvio‎u sly had had a diffe‎r ent opini‎o n on how the eveni‎n g had gone. When we got home that night‎, after‎we put Shosh‎a na to bed, he said modes‎t ly, "Well, I think‎we hit it off A-OK."Words‎: 792。

合同书的英语翻译

合同书的英语翻译Contract TranslationThis Contract (“Contract”) is entered into by and between [Party A] and [Party B], which sets forth the rights and obligations of both parties and their respective performances, as well as the terms applicable to this Contract. This Contract is in compliance with relevant laws and regulations of China.Article 1. Basic Information1.1 [Party A]Name: [insert name]Address: [insert address]Legal representative: [insert name]Contact information: [insert phone number and email]1.2 [Party B]Name: [insert name]Address: [insert address]Legal representative: [insert name]Contact information: [insert phone number and email]Article 2. Identities and Responsibilities2.1 [Party A] is a [insert identity] and is responsible for [insert responsibilities].2.2 [Party B] is a [insert identity] and is responsible for [insert responsibilities].Article 3. Rights and Obligations3.1 [Party A] has the following rights and obligations:[insert rights and obligations]3.2 [Party B] has the following rights and obligations:[insert rights and obligations]Article 4. Performance and Term4.1 [Party A] will perform [insert performance] according to this Contract and the relevant laws and regulations of China.4.2 [Party B] will perform [insert performance] according to this Contract and the relevant laws and regulations of China.4.3 The term of this Contract is [insert term].Article 5. Breach of Contract5.1 Any breach of the terms of this Contract by either party will result in a breach of contract.5.2 The party in breach of contract shall assume full responsibility for any losses incurred by the other party.Article 6. Applicable Laws6.1 This Contract shall be governed by and construed in accordance with the laws of China.6.2 Any dispute arising from or in connection with this Contract shall be settled through friendly negotiation. If the parties fail to reach an agreement through negotiation, the dispute shall be submitted to the [insert dispute resolution organization] for arbitration.Article 7. Legal Force and Enforceability7.1 This Contract shall be legally binding on both parties.7.2 This Contract is enforceable in accordance with the applicable laws and regulations of China.Article 8. Other Provisions[Insert any other provisions as needed]This Contract is executed in duplicate, with each party retaining one copy. Both parties hereby acknowledge their understanding and agreement with the terms of this Contract, as well as their respective rights and obligations.[Party A]Signature: [insert signature]Date: [insert date][Party B]Signature: [insert signature]Date: [insert date]。

未能达成协议英语

未能达成协议英语English Version of Failed AgreementThis document serves as an agreement that was not reached between the involved parties, and outlines the basic information for each party, their identities, rights, obligations, performance, deadlines, and breach of responsibilities. The agreement also states that the parties involved must comply with China's relevant laws and regulations, outlines the rights and responsibilities of each party, and clearly delineates the legal effectiveness and enforceability of these terms. All clauses in this document comply with the legal requirements.Party A: [insert basic information]Party B: [insert basic information]Identification of Parties:Party A is identified as [insert identity information], and party B is identified as [insert identity information]Rights and Obligations:1. Both parties agree to negotiate in good faith and make every effort to reach a mutually satisfactory agreement.2. Each party must abide by the terms of the agreement as stated in this document.3. Party A is responsible for [insert obligation], and party B is responsible for [insert obligation].4. [Insert other pertinent rights and obligations]Performance:1. Party A must [outline performance obligation], and party B must [outline performance obligation].2. If either party fails to fulfill their obligations, the other party is entitled to take legal actions to seek redress.Deadline:1. The agreement is valid from the date of signature by both parties and shall remain binding.2. [Insert applicable deadline for performance]Breach of Responsibility:1. If either party breaches the agreement or fails to perform their obligations, they must compensate the other party for any resulting losses.2. Any dispute arising from this agreement will be settled through consultation between the parties. If consultation does not result in an acceptable outcome, both parties agree to submit the dispute to the relevant legal authorities for resolution.Compliance:1. This agreement shall be carried out in compliance wi th China’s relevant laws and regulations.2. Both parties agree to adhere to all applicable Chinese laws, regulations, and policies.Legal Effectiveness and Enforceability:1. Each party warrants that they have the authority and capacity to enter into this agreement and that all information provided is accurate and complete.2. This document is legally binding on both parties and shall be enforceable according to Chinese law.3. Any changes to the agreement must be made in writing and signed by both parties.This agreement is written in English. Party A:[insert signature] Date:[insert] Party B:[insert signature] Date:[insert]。

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a r X i v :m a t h /0412015v 3 [m a t h .C O ] 15 D e c 2004A note on two identities arising from enumeration of convexpolyominoesVictor J.W.Guo 1and Jiang Zeng 1,21Center for Combinatorics,LPMCNankai University,Tianjin 300071,People’s Republic of Chinajwguo@2Institut Girard Desargues,Universit´e Claude Bernard (Lyon I)F-69622,Villeurbanne Cedex,France zeng@desargues.univ-lyon1.frAbstract.Motivated by some binomial coeﬃcients identities encountered in our approach to theenumeration of convex polyominoes,we prove some more general identities of the same type,one of which turns out to be related to a strange evaluation of 3F 2of Gessel and Stanton.Keywords :binomial coeﬃcients identities,generating function AMS Classiﬁcation :05A15,05A191IntroductionIn our elementary approach to the enumeration of convex polyominoes with an (m +1)×(n +1)minimal bounding rectangle [5],we encountered the following two interesting identities:m a =1n b =1m +n −a +b −1m −a m +n +a −b −1n −b=mn2(m +n ) m +nm2−2m +2n 2n.(2)Although the single-sum case of binomial coeﬃcients identities is well-studied,the symbolic manipulation of binomial multiple-sum identities depends on the performance of computers (see,for example,[2]).Therefore,formulas of binomial double-sums are still a challenge both for human and computer.In this paper,we will give some generalizations of the above two formulas,our main results are the following two theorems.1Theorem 1For m,n ∈N and any number α=0,we havem a =1n b =1(1+α)m −a +b −1m −a (1+α−1)n +a −b −1n −b=mn(1+α)(m +α−1n )(1+α)m m (1+α−1)nn+r k =−r(r −|k |+1) (1+α)m m −k (1+α−1)nn −k.(4)We use essentially the generating function techniques,that is,to prove A =B we show thattheir generating functions are equal.Two proofs of Theorem 1will be given in Section 1,and Theorem 2will be proved in Section 3.In Section 4,we derive some interesting special cases of Theorems 1and 2.2Proofs of Theorem 12.1First Proof of Theorem 1.Multiplying the left-hand side of (3)by x m y n and summing over m ≥0and n ≥0we obtain the generating function F (x,y ),which can be written after exchanging the order of summations as:F (x,y ):=∞ a,b =1x a y b ∞ m,n =0(1+α)m +aα+b −1m (1+α−1)n +a +bα−1−1n x m y n .Summing the two inner sums by the following classical formula (see [6,p.146])and [4,(9)]):∞ n =0 α+βn n w n =z α+1z β,and then summing the two resulted geometric series over a and b we obtainF (x,y )=uv (u −1)(v −1)wherex=u−1v1+α−1.(5)Now,using the fact that k≥0kx k=x/(1−x)2we haveF(x,y)=uv(1+α)(m+α−1n) (1+α)m m (1+α−1)n n .(6)Writingk (1+α)m m−k (1+α−1)n n−k =(m+α−1k)(n+αk)(1+α)(m+α−1n) (1+α)mm−k−1 (1+α−1)nn−k−1 , equation(6)follows then by summing over k from0to min{m,n}.1+α−αu ,∞n=0 (1+α−1)n n x n=v∂x +(1+α−1)y∂dx u dyvApplying the transformation [1,p.142]:3F 2−N,a,b d,e;1 =(e −b )N(−(1+α−1)n −a +2)n −1·3F 21−n,(1+α)m +1−a,αm αm +1,(1+α)m +α−1n +1;1.Expanding the 3F 2as a k -sum and exchanging the order with a -sum yieldsL =(1+α)m m (1+α−1)n −2n −1(−(1+α)(m +α−1n )+1)n −1(αm +1)k ((1+α)m +α−1n +1)k k !m(2−(1+α−1)n )n −1·m(1+s (b +N ))N (b +N ),with N =n −1,b =αm +1and s =−1−α−1.2)Γ(1+aΓ(1+a )Γ(1+a2−c )Γ(1+a −b −c ),and if m =n ,we can apply Whipple’s formula [1,p.149]:3F 2a,1−a,cd,1+2c −d;1=21−2c πΓ(d )Γ(1−2c +d )2+a2)Γ(a 2)Γ(1−a 2)Γ(12+d3Proof of Theorem2 Consider the generating function:G r(x,y):=∞m,n=−r m+r a=0n+r b=0 (1+α)m−a+b−1m+r−a (1+α−1)n+a−b−1n+r−b x m y n.Using(5),as theﬁrst proof of Theorem1,we haveG r(x,y)=uv(u−1)−r(v−1)−r(1+α−αu)(1+α−1−α−1v)(uv−u−v)2.On the other hand,for−r≤k≤r,we have∞m,n=−r (1+α)m m−k (1+α−1)n n−k x m y n=uv(u−1)k(v−1)k(1+α−αu)(1+α−1−α−1v)(uv−u−v)2+rk=−r(r−|k|+1)uv(u−1)k(v−1)k4Some consequences4.1Consequences of Theorem1Replacingα,m,and n by q/p,pm,and qn,respectively,in Theorem1,we obtain Corollary3For positive integers m,n,p,and q,there holdspma=1qn b=1 pm+qm−a+b−1pm−a pn+qn+a−b−1qn−b=pqmnExchanging p and m ,and q and n ,respectively,Corollary 3may be written as follows:pm a =1qn b =1pm +qm −a +b −1b −1 pn +qn +a −b −1a −1 =pqmnp +qpm +qm +pn +qnpm +pn.Therefore,by Corollary 3,we have0 a =1−pn qn b =1pm +qm −a +b −1pm −a pn +qn +a −b −1qn −b =pqn(p +q )(m +n )pm +qm pm pn +qnpn.Replacing a by 1−a ,we obtainpn a =1qn b =1pm +qm +a +b −2pm +a −1 pn +qn −a −bqn −b =pqn(p +q )(m +n ) pm +qm pm pn +qn pn.Dividing both sides by pm +qm pm,we get pn a =1qn b =1(pm +qm +1)a +b −2(p +q )(pm +1)pn (qm +1)qn−pqmn(mx +1)a −1(nx +1)b −1 m +n −a −bm −a=mn (mx +nx +1)m +n(m +n )(1+x ) m +nm.6Letting x −→∞,we obtainCorollary 5For m,n ∈N ,there holdsm a =1n b =1m +n −a −b m −a (m +n )a +b −2m m −1n n −1−mn(p +q )(m +n )pm +qm pm pn +qnpn+r k =−r(r −|k |+1) pm +qm pm −k pn +qnqn −k.(11)Namely,pm −r −1 a =1qn −r −1 b =1pm +qm −a +b −1pm −r −1−a pn +qn +a −b −1qn −r −1−b+0 a =−pm −r 0b =−qn −rpm +qm +a −b −1pm +r +a pn +qn −a +b −1qn +r +b=2pqmnp +q(p +q )(m +n )pm +pn,(13)7anda =−pm −rqm −r −1 b =1pm +qm +a −b −1pm +r +a pn +qn −a +b −1qn +r +bpn −r −1 a =1qm −r −1 b =1pm +qm +a −b −1pm +r +apn +qn −a +b −1qn +r +b=(qm −r −1)p(p +q )(m +n )pm +qm pm pn +qn pn −r(p +q )(m +n )pm +pn+r −1 k =1−r(r −|k |) pm +qm pm −k pn +qnqn −k.(15)And for the case r is negative,a similar formula can be deduced from(11).For the p =q =1and m =n =1cases,we obtain the following two corollaries:Corollary 7For positive integers m ,n ,and r ,we havem −r a =1n −r b =12m −a +b −1m −r −a 2n +a −b −1n −r −b=mn22m +2n m +n+r2(m +n )m +n m2−r2m +n m 2+r −1k =1(r −k ) m +n m −k m +nn −k.8Furthermore,when r =1and r =2we obtain the following:m −2 a =1n −2 b =12m −a +b −1m −a −2 2n +a −b −1n −b −2 = 2m m 2n n + 2m m −1 2n n −1+mn 2 2mm2nn+mn22m +2nm +n,m −1 a =1n −1 b =1m +n −a +b −1m −a −1 m +n +a −b −1n −b −1=12(m +n )m +nm2−12x +m2x +m2n.Proof.Replacing x by −x −m +n ,one sees that the theorem is equivalent to2m a =1n b =1x +a −1n +b −1 x +m −a n −b =m 2x +m −12n −1.(16)Now,changing a to m +1−a and b to 1−b ,respectively,we obtainm a =1n b =1x +a −1n +b −1 x +m −a n −b =m a =10 b =1−nx +m −a n −b x +a −1n +b −1.So we can rewrite the left-hand side of (16)as follows:m a =1n b =1x +a −1n +b −1 x +m −a n −b +m a =10 b =1−nx +m −a n −b x +a −1n +b −1=m a =1n b =1−nx +a −1n +b −1 x +m −an −b=m2x +m −12n,where the last step follows from Chu-Vandermonde’s formula.AcknowledgementWe thank Christian Krattenthaler for his helpful comments,which,in particular,led to the second proof of Theorem1.References[1]G.Andrews,R.Askey and R.Roy,Special Functions,Encyclopedia of Math.and its applications,Vol.71,Cambridge University Press,Cambridge,UK,1999.[2]G.Andrews and P.Paule,Some questions concerning computer-generated proofs of a binomialdouble-sum identity,J.Symbolic Comput.16(1993),147–153.[3]I.Gessel and D.Stanton,Strange evaluations of hypergeometric series,SIAM.J.Math.Anal.13(2)(1982),295-308.[4]H.W.Gould,Some generalizations of Vandermonde’s convolution,Amer.Math.Monthly63(1956)84–91.[5]V.J.W.Guo and J.Zeng,The number of convex polyominoes and the generating function of Jacobipolynomials,available at 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