Dedicated to the memory of Gian-Carlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the η-function identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing

The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1 / ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdös in 1948. We describe a formally verified version of Selberg’s proof, obtained using the Isabelle proof assistant. 1

Abstract. We describe a formalization of asymptotic O notation using the Isabelle/HOL proof assistant. 1

Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of first-order arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1

Dedicated to Professor Bahman Mehri on the occasion of his 70th birthday ABSTRACT. In this paper we study vp(n!), the greatest power of prime p in factorization of n!. We find some lower and upper bounds for vp(n!), and we show that vp(n!) = n p−1 + O(ln n). By using the afore mentioned bounds, we study the equation vp(n!) = v for a fixed positive integer v. Also, we study the triangle inequality about vp(n!), and show that the inequality pvp(n!)> qvq(n!) holds for primes p < q and sufficiently large values of n.

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Abstract--This paper presents an enhanced Latin Square Genetic Algorithm (LSGA). It makes the chromosomes to be more sensible to their surrounding regions. The algorithm applies orthogonal design method to every chromosome in the population to detect chromosomes with high fitness values in the surrounding regions. Orthogonal design method makes it more concise and direct to find the delegate to represent the situation of the surrounding regions. We execute the proposed algorithm to solve 15 test functions and compare it with traditional algorithm without using orthogonal design method. The results show that the proposed algorithm can find optimal or close-to-optimal solutions with higher speed and more accuracy.

Abstract. We study the closure in the Hardy space or the disk algebra of algebras generated by two bounded functions, of which one is a finite Blaschke product. We give necessary and sufficient conditions for density or finite codimension (of the closure) of such algebras. The conditions are expressed in terms of the inner part of some function which is explicitly derived from each pair of generators. Our results are based on identifying z-invariant subspaces included in the closure of the algebra. 1.

ektivn, dv konkrtn funkci f : N ! N takovou, e mnoina f1; 2; : : : ; f(k)g pro kad k obsahuje aritmetickou posloupnost dlky k sloenou z prvosel. Tao v [49] uvd, e lze vzt f(k) = 2 2 2 2 2 100k Green a Tao modi kac dkazu vty 1.1 dokzali jej zeslen, vtu 1.2: Je-li P mnoina vech prvosel a podmnoina Q P spluje lim sup n!1 Q(n) P (n) = c > 0 (Q(n) je poet prvk q v Q, q n, podobn P (n)), mus Q obsahovat libovoln dlouh aritmetick posloupnosti. Je znmo, e pro Q 1 = fp 2 P : p = 4n+1g mme c = 1=2 a kad p 2 Q 1 je souet dvou tverc (p = a +b pro dv pirozen sla a; b, viz st 3). Vta 1.2 tedy dv (napklad) dosud neznm fakt, e existuj libovoln dlouh aritmetick posloupnosti tvoen souty dvou tverc. (Nap. 37 = 1 , 61 = 5 , 85 = 9 + 2 , 109 = 10 + 3 je takov posloupnost.) 2 Dkaz Greenovy a Taovy vty o prvoslech Pirozen sla f1; 2; : : :g ozname N a mnoinu f1; 2; : : : ; Ng, pro N 2 N, jako [N ]. Symboly Z, Q, R a C oznauj mnoiny celch, racionlnch, relnch a komplex

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