Datasets:

Asilarkness
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DiffuRefill-facts

+{"text": "A video game may be regarded as a discrete dynamical system that evolves in response to inputs supplied by a player. Formally, let \\(S\\) denote the (finite) set of all possible configurations of the game, each element of which is called a state. A state records the complete information required to resume the game at any instant: the positions of all characters, the values of all counters, the contents of the memory, and any other data that influences future evolution. The set \\(S\\) is typically very large, but it is nonetheless a subset of the Cartesian product of a finite number of integer intervals, so that each state can be encoded as a finite string of bits. The player’s actions are modeled by a set \\(A\\) of admissible actions; each action is identified with a natural number, for instance by enumerating the buttons on a controller as \\(0,1,\\dots ,k-1\\). The interaction between player and game is captured by a transition function\n\\[\nT\\colon S\\times A\\longrightarrow S,\n\\]\nwhich assigns to each pair \\((s,a)\\) a unique successor state \\(s' = T(s,a)\\). The function \\(T\\) embodies the game’s rules: it specifies how the world reacts when a particular button is pressed while the game is in a given configuration.\n\nTo illustrate these ideas, consider a simple platformer in which the player controls a character that can either stand still, move one unit to the right, or jump. Let the horizontal coordinate of the character be an integer \\(x\\in\\mathbb N\\) and let the vertical coordinate be either \\(0\\) (on the ground) or \\(1\\) (in the air). The state space is therefore\n\\[\nS=\\{(x,y)\\mid x\\in\\mathbb N,\\; y\\in\\{0,1\\}\\}.\n\\]\nWe enumerate the three possible actions as \\(a_{0}= \\text{stay}\\), \\(a_{1}= \\text{right}\\), and \\(a_{2}= \\text{jump}\\). The transition function is defined by\n\\[\nT((x,0),a_{0})=(x,0),\\qquad\nT((x,0),a_{1})=(x+1,0),\\qquad\nT((x,0),a_{2})=(x,1),\n\\]\nand, because the character cannot jump while airborne, we set\n\\[\nT((x,1),a_{0})=(x,0),\\qquad\nT((x,1),a_{1})=(x+1,0),\\qquad\nT((x,1),a_{2})=(x,1).\n\\]\nThe rule \\(T((x,1),a_{0})=(x,0)\\) models the effect of gravity: after one time step in the air the character returns to the ground. Starting from the initial state \\(s_{0}=(0,0)\\), a sequence of actions \\((a_{2},a_{1},a_{0},a_{1})\\) yields the state trajectory\n\\[\n(0,0)\\xrightarrow{a_{2}}(0,1)\\xrightarrow{a_{1}}(1,0)\\xrightarrow{a_{0}}(1,0)\\xrightarrow{a_{1}}(2,0).\n\\]\nEach step is computed by applying \\(T\\) to the current state and the next action. In this manner the entire play of the game can be described as a finite or infinite sequence\n\\[\ns_{0},s_{1},s_{2},\\dots ,\\qquad s_{i+1}=T(s_{i},a_{i}),\n\\]\nwhere the \\(a_{i}\\) are chosen by the player. The rigorous formulation above makes clear that a video game is nothing more than a deterministic state‑transition system driven by a stream of natural‑number‑encoded inputs, and it provides the foundation for analysis, testing, and formal verification of interactive software."}
+{"text": "Antonino Fernández Rodríguez (1917‑2016) is remembered principally as the architect of one of the most diversified corporate groups in Latin America, Grupo Cisneros, whose origins lie in a modest import‑export firm founded in Caracas in 1929. Fernández’s business philosophy can be distilled into three interlocking concepts: vertical integration, strategic diversification, and the disciplined use of capital as a lever for sustainable growth. Each of these ideas can be illustrated with a simple numerical model that uses only natural numbers, thereby showing how the abstract principles translate into concrete decision‑making.\n\nConsider a hypothetical subsidiary of Cisneros that imports a commodity, processes it, and sells the finished product domestically. Let the natural number \\(n\\) denote the number of production stages the firm decides to add in a given planning horizon. The firm begins with a single stage (\\(n=1\\)) in which it merely imports and resells the raw material, earning a profit \\(p_{1}=5\\) million bolívares. Vertical integration is the decision to increase \\(n\\) by adding a processing stage, thereby capturing the margin that would otherwise belong to an external processor. If the firm adds one processing stage (\\(n=2\\)), the additional value created is represented by a constant marginal profit increment \\(\\Delta p=3\\) million bolívares per stage, so the total profit becomes \\(p_{2}=p_{1}+ \\Delta p =8\\) million. The firm may continue to integrate further; for \\(n=3\\) the profit is \\(p_{3}=p_{2}+ \\Delta p =11\\) million, and so on. The profit function is therefore linear in the natural number of stages:\n\\[\np(n)=5+3\\,(n-1),\\qquad n\\in\\mathbb{N},\\; n\\ge1 .\n\\]\n\nStrategic diversification enters when the firm evaluates the opportunity cost of allocating capital to a new, unrelated line of business. Suppose the firm has a capital budget of \\(C=30\\) million bolívares and each additional stage costs a fixed amount \\(c=6\\) million. The feasible set of stage numbers is the set of natural numbers \\(n\\) satisfying \\(c\\,(n-1)\\le C\\), i.e. \\(n\\le 6\\). The profit maximization problem is then to choose \\(n\\) in \\(\\{1,2,3,4,5,6\\}\\) that maximizes \\(p(n)\\). Because \\(p(n)\\) is strictly increasing, the optimal choice is \\(n^{*}=6\\), yielding a profit of \\(p(6)=5+3\\cdot5=20\\) million. The residual capital, \\(C- c\\,(n^{*}-1)=30-6\\cdot5=0\\), can be redeployed in a distinct sector—say media—illustrating diversification.\n\nFinally, Fernández’s disciplined capital management is captured by the requirement that the incremental profit \\(\\Delta p\\) must exceed the incremental cost \\(c\\) for any added stage; otherwise the firm would halt integration. In the example, \\(\\Delta p=3\\) million is less than \\(c=6\\) million, so a realistic model would impose a constraint \\(\\Delta p\\ge c\\) to reflect Fernández’s prudence. Adjusting the parameters to \\(\\Delta p=7\\) million restores the inequality, making each additional stage profitable and justifying the vertical integration that characterized Fernández’s expansion of Cisneros from a single import house to a conglomerate spanning telecommunications, media, and consumer goods. This compact numerical illustration therefore encapsulates the essence of Antonino Fernández Rodríguez’s enduring business doctrines."}
+{"text": "The RSA‑CRT‑based key generation method that was widely used in smart‑card firmware and in some cryptographic libraries relied on the construction of a prime of the form  \n\n\\[\np = k\\;2^{t}+1 ,\n\\]\n\nwhere the integer \\(t\\) is a public parameter chosen by the implementer and the secret integer \\(k\\) is drawn uniformly from a small interval.  The corresponding prime \\(q\\) is obtained in the same way with a different secret multiplier.  The public modulus is then \\(N=p\\,q\\).  This scheme was introduced in order to speed the generation of strong primes, because testing candidates of the form \\(k2^{t}+1\\) for primality is cheaper than testing arbitrary odd numbers.  The vulnerability, known as ROCA (Return of Coppersmith’s Attack), arises because the structure of \\(p\\) and \\(q\\) leaks information about the secret multipliers \\(k_{p}\\) and \\(k_{q}\\), and consequently about the private exponent \\(d\\).  The attack exploits the fact that the set of admissible primes is a lattice of relatively low dimension; Coppersmith’s method for finding small roots of modular polynomial equations can be applied to recover the hidden multipliers and thus factor the modulus.\n\nTo see the mathematics, let us fix the public parameter \\(t=16\\).  Suppose a device produces a prime  \n\n\\[\np = 3\\cdot 2^{16}+1 = 3\\cdot 65536+1 = 196609 .\n\\]\n\nA quick primality test confirms that \\(p\\) is indeed prime.  The second prime is generated with a secret multiplier \\(k_{q}=7\\), giving  \n\n\\[\nq = 7\\cdot 2^{16}+1 = 7\\cdot 65536+1 = 458753 .\n\\]\n\nThe public modulus is  \n\n\\[\nN = p\\,q = 196609 \\times 458753 = 90\\,221\\,224\\,577 .\n\\]\n\nAn attacker who knows only \\(N\\) and the public exponent \\(e\\) (for instance \\(e=65537\\)) wishes to recover the private key \\(d\\).  Because both factors share the same power‑of‑two offset, they satisfy  \n\n\\[\np \\equiv 1 \\pmod{2^{t}}, \\qquad q \\equiv 1 \\pmod{2^{t}} .\n\\]\n\nConsequently  \n\n\\[\nN \\equiv 1 \\pmod{2^{t}} .\n\\]\n\nIndeed, \\(N \\bmod 2^{16}=1\\).  The attacker forms the polynomial  \n\n\\[\nf(x)=x\\;2^{t}+1,\n\\]\n\nand seeks a small integer root \\(x\\) such that \\(f(x)\\) divides \\(N\\).  Coppersmith’s theorem guarantees that any root \\(x_{0}\\) with \\(|x_{0}|<N^{1/4}\\) can be found in polynomial time from the congruence  \n\n\\[\nf(x) \\equiv 0 \\pmod{N}.\n\\]\n\nIn our example \\(N^{1/4}\\approx 562\\), and the secret multiplier for \\(p\\) is \\(k_{p}=3\\), which is well below this bound.  By constructing a lattice basis from the coefficients of \\(f(x)\\) and applying the LLL reduction algorithm, the attacker obtains a short vector that reveals the integer \\(k_{p}=3\\).  Once \\(k_{p}\\) is known, the factor \\(p\\) is recovered as \\(p=k_{p}2^{t}+1\\).  Dividing \\(N\\) by \\(p\\) yields \\(q\\), and the private exponent \\(d\\) follows from the relation \\(d\\equiv e^{-1}\\pmod{(p-1)(q-1)}\\).\n\nThe essential insight of the ROCA attack is that the secret multipliers are bounded by a value much smaller than \\(\\sqrt{N}\\); therefore they constitute “small roots’’ of a modular equation.  Coppersmith’s method turns this observation into an algorithm that factors any RSA modulus generated with the vulnerable prime‑generation recipe, provided the public parameter \\(t\\) is known.  The attack is deterministic, runs in a few seconds on a modern laptop, and invalidates all keys produced by the affected firmware, prompting the deprecation of the prime‑generation formula in favor of truly random strong primes."}
+{"text": "The monumental ensemble known as Worker and Kolkhoz Woman was conceived in the early 1930s as the central feature of the Soviet exhibition “World of Labor” held in Moscow’s Gorky Park. Its creators, the sculptors Vladimir Zav’yalov and Matvey Manizer, were instructed by the state to produce a work that would embody the official ideology of socialist realism: the glorification of the proletariat and the collective farm as twin pillars of the communist future. In technical terms the sculpture is a bronze composition, 24 meters high, supported on a steel frame that distributes its weight uniformly across a concrete pedestal of 2 meters in height. The two figures stand on a shared pedestal, each holding a hammer and a sickle, the universal symbols of industry and agriculture. The composition is deliberately symmetrical; the axis of symmetry passes through the vertical line that joins the heads of the two figures and the centre of the pedestal.\n\nTo understand the engineering constraints that the state‑mandated dimensions impose, consider a worked example in which the total mass of the bronze is required to be an integer multiple of a basic “state unit” of mass, defined by the government as 1 tonne = 10³ kg. Let the density of the bronze alloy be ρ = 8 700 kg · m⁻³. The volume V of the bronze is therefore V = M/ρ, where M is the total mass. The designers chose M = 88 tonnes, a natural number that is also a multiple of 8, the number of “five‑year plans” completed by the time of the sculpture’s unveiling. Substituting, V = 88 × 10³ kg ⁄ 8 700 kg · m⁻³ ≈ 10.115 m³. The volume is then distributed between the two figures in the ratio 7:5, reflecting the state’s emphasis on industrial workers (7 parts) over collective farm women (5 parts). Thus the worker’s bronze volume is (7⁄12) × 10.115 ≈ 5.90 m³, and the woman’s volume is (5⁄12) × 10.115 ≈ 4.21 m³. These volumes correspond to masses of 5.90 × 8 700 ≈ 51.3 tonnes and 4.21 × 8 700 ≈ 36.7 tonnes respectively, both of which round to the nearest natural number of tonnes (51 t and 37 t) as required by the state’s reporting standards.\n\nThe symbolism of the numbers mirrors the ideological narrative: the total mass of 88 tonnes echoes the year 1888, the birth year of several leading Bolsheviks, while the ratio 7:5 encodes the proportion of industrial to agricultural output that Soviet planners projected for the forthcoming decade. By embedding such precise natural‑number relationships within the physical parameters of the work, the artists fulfilled the state’s demand for a sculpture that was not only visually striking but also mathematically resonant with the official discourse of progress. The result is a seamless fusion of art, engineering, and political semiotics that continues to dominate the skyline of Moscow and to illustrate the manner in which Soviet cultural production was calibrated to the exacting standards of the planned economy."}
+{"text": "Let \\(F\\) be a free group on a set \\(X\\).  By definition there is a group homomorphism\n\\(\\iota\\colon X\\to F\\) such that for every group \\(G\\) and every map\n\\(f\\colon X\\to G\\) there exists a unique homomorphism \\(\\widehat f\\colon F\\to G\\) with\n\\(\\widehat f\\circ\\iota=f\\).  The universal property forces the elements of \\(F\\) to be\nrepresented uniquely by reduced words in the alphabet\n\\(X^{\\pm 1}=X\\cup X^{-1}\\).  A **word** is a finite sequence\n\\(w=x_{1}^{\\varepsilon _1}x_{2}^{\\varepsilon _2}\\cdots x_{n}^{\\varepsilon _n}\\) with\n\\(x_i\\in X\\) and \\(\\varepsilon_i\\in\\{+1,-1\\}\\).  A word is **reduced** if no\nconsecutive pair \\(x_i^{\\varepsilon_i}x_{i+1}^{\\varepsilon_{i+1}}\\) satisfies\n\\(x_i=x_{i+1}\\) and \\(\\varepsilon_i=-\\varepsilon_{i+1}\\); that is, no letter is\nimmediately followed by its inverse.  Two words represent the same element of\n\\(F\\) precisely when one can be transformed into the other by a finite sequence\nof cancellations of adjacent inverse pairs.  The **normal form theorem for free\ngroups** states that every element of \\(F\\) can be written uniquely as a reduced\nword, and that the empty word represents the identity.  Consequently the map\nthat sends a reduced word to its class in \\(F\\) is a bijection; the group\noperation is concatenation followed by reduction.\n\nThe same idea extends to the **free product** of a family of groups\n\\(\\{G_i\\}_{i\\in I}\\).  Denote the free product by \\(\\ast_{i\\in I} G_i\\).  Formally it\nis the group equipped with injections \\(\\iota_i\\colon G_i\\to\\ast_{i\\in I}G_i\\) that\nsatisfy the universal property: for any group \\(H\\) and homomorphisms\n\\(\\phi_i\\colon G_i\\to H\\) there exists a unique homomorphism\n\\(\\Phi\\colon\\ast_{i\\in I}G_i\\to H\\) with \\(\\Phi\\circ\\iota_i=\\phi_i\\) for all \\(i\\).\nElements of \\(\\ast_{i\\in I}G_i\\) admit a normal form analogous to the reduced\nwords of a free group.  A **syllable** is a non‑identity element of some factor\n\\(G_i\\).  A **reduced word** (or normal form) is a finite product\n\\(g_{1}g_{2}\\cdots g_{k}\\) where each \\(g_j\\in G_{i_j}\\setminus\\{1\\}\\) and the\nindices satisfy \\(i_j\\neq i_{j+1}\\) for all \\(j\\).  Two reduced words represent\nthe same element iff they are identical; the empty product is the identity.\nThe normal form theorem for free products asserts precisely this uniqueness.\n\nConsider a concrete example.  Let \\(G_1=\\langle a\\mid a^2=1\\rangle\\cong C_2\\) and\n\\(G_2=\\langle b\\mid b^3=1\\rangle\\cong C_3\\).  Their free product\n\\(G=G_1\\ast G_2\\) consists of all reduced words built from the non‑trivial\nelements \\(a\\) (order two) and \\(b,b^2\\) (order three), with the rule that no two\nconsecutive syllables come from the same factor.  For instance,\n\\(w=ab^2ab\\) is reduced: the syllables alternate between \\(G_1\\) and \\(G_2\\) and\nnone equals the identity.  Multiplying \\(w\\) by the element \\(b\\) on the right\ngives \\(ab^2ab\\,b\\).  Since the last two syllables both lie in \\(G_2\\), we first\nconcatenate them to obtain \\(ab^2a\\,b^3\\); but \\(b^3=1\\) in \\(G_2\\), so the final\nsyllable disappears and the product reduces to the word \\(ab^2a\\), which is\nalready reduced because the adjacent syllables belong to different factors.\nThus the normal form provides a systematic way to compute in the free product\nand guarantees that each group element has a unique description as a reduced\nalternating product of non‑trivial factor elements."}
+{"text": "The quantity called **jerk** (sometimes denoted \\(j\\) or \\(\\dot a\\)) is the third‑order time derivative of the position vector of a particle moving in Euclidean space. If \\(\\mathbf r(t)\\) gives the location of the particle at time \\(t\\), the velocity is \\(\\mathbf v(t)=\\frac{d\\mathbf r}{dt}\\), the acceleration is \\(\\mathbf a(t)=\\frac{d\\mathbf v}{dt}= \\frac{d^{2}\\mathbf r}{dt^{2}}\\), and the jerk is defined precisely by  \n\n\\[\n\\mathbf j(t)=\\frac{d\\mathbf a}{dt}= \\frac{d^{3}\\mathbf r}{dt^{3}} .\n\\]\n\nThus jerk measures the rate at which the acceleration itself changes with time. In a one‑dimensional setting the scalar jerk is \\(j(t)=\\frac{d^{3}x}{dt^{3}}\\) where \\(x(t)\\) is the position coordinate. The SI unit of jerk is metres per second cubed \\(\\mathrm{m\\,s^{-3}}\\).\n\nJerk is not merely a mathematical curiosity; it appears in the design of motion‑control systems, vehicle suspension engineering, and human‑perceived comfort. A sudden change in acceleration (large jerk) can cause mechanical stress or discomfort, whereas a smooth motion profile typically limits jerk to a prescribed bound.\n\nTo illustrate the concept, consider a particle that moves along a straight line with a position function  \n\n\\[\nx(t)=\\frac{1}{6}t^{3}+t^{2}+2t .\n\\]\n\nThe first derivative gives the velocity  \n\n\\[\nv(t)=\\frac{dx}{dt}= \\frac{1}{2}t^{2}+2t+2 .\n\\]\n\nDifferentiating once more yields the acceleration  \n\n\\[\na(t)=\\frac{dv}{dt}= t+2 .\n\\]\n\nA third differentiation provides the jerk  \n\n\\[\nj(t)=\\frac{da}{dt}=1 .\n\\]\n\nHence the jerk is constant and equal to \\(1\\;\\mathrm{m\\,s^{-3}}\\) for all \\(t\\). At \\(t=0\\) the particle starts from the origin with zero velocity, an acceleration of \\(2\\;\\mathrm{m\\,s^{-2}}\\), and a jerk of \\(1\\;\\mathrm{m\\,s^{-3}}\\). After one second the acceleration has increased to \\(3\\;\\mathrm{m\\,s^{-2}}\\) because the constant jerk has added \\(1\\;\\mathrm{m\\,s^{-2}}\\) per second to the acceleration. After three seconds the acceleration reaches \\(5\\;\\mathrm{m\\,s^{-2}}\\) and the velocity has grown to \\(v(3)=\\frac{1}{2}(9)+6+2=15.5\\;\\mathrm{m\\,s^{-1}}\\).\n\nIf the jerk were zero, the acceleration would remain constant and the motion would be uniformly accelerated. Conversely, a non‑zero jerk implies that the acceleration is not constant; the sign of the jerk indicates whether the acceleration is increasing (\\(j>0\\)) or decreasing (\\(j<0\\)). In engineering practice one often specifies a maximum allowable jerk, \\(j_{\\max}\\), and designs control laws that keep \\(|j(t)|\\le j_{\\max}\\) to avoid excessive forces on mechanical components or passengers. The mathematical definition of jerk thus provides a precise tool for quantifying and controlling the smoothness of motion in both theoretical analysis and practical applications."}