A new proof of the existence of (q 2−q+1)-arcs inPG(2,q 2)

Journal of Geometry -     

On the neutrix convolution product ofx − r Inx − andx + −3

The existence of the neutrix convolution product of distributionx _? ^r Inx _? andx ₊ ^?3 is proved and some convolution products are evaluated.     

On the diophantine equation X 2 − (1 + a 2)Y 4 = −2a

Let a≥1 be an integer.In this paper,we will prove the equation in the title has at most three positive integer solutions.     

On the left side derivative of the Hausdorff dimension of the Julia sets for z 2 + c at c = −3/4

Let d(c) denote the Hausdorff dimension of the Julia set J _c of the polynomial f _c (z) = z ² + c. The function c ? d(c) is real-analytic on the interval (?5/4,?3/4), which is included in the 1/2 bulb of the Mandelbrot set. The number c = ?3/4 is the parameter at which, going from the right, the attracting fixed point bifurcates to an orbit of period two. Recently [13], we studied d′(c) when c ? ?3/4 from the right. Here, under numerically verified assumption d(?3/4) < 4/3, we will show that there exists K _?3 ^4/? > 0 such that d′(c)(?3/4 ? c)^{?3d(?3/4)/2+2} → ?K _?3 ^4/? , when c tends to ?3/4 from the left. In particular we obtain d′(c) ? ?∞. This case is much harder than that considered in [13].     

Characterization of {(q + 1) + 2, 1;t,q}-min · hypers and {2(q + 1) + 2, 2; 2,q}-min · hypers in a Finite projective geometry

In this paper, we shall characterize all {(q + 1) + 2, 1;t, q}-min · hypers and all {2(q + 1) + 2, 2; 2,q}-min · hypers for any integert 2 and any prime powerq 3. In the next paper [8], we shall characterize all {2(q + 1) + 2, 2;t, q}-min · hypers for any integert 3 and any prime powerq 5 using the results in this paper.     

On m-regular systems on ℋ(5,q 2)

The notion of m-regular system on the Hermitian variety ℋ(n,q ²) was introduced by B. Segre (Ann. Math. Pura Appl. 70:1–201, 1965). Here, three infinite families of hemisystems on ℋ(5,q ²), q odd, are constructed.     

On the Diophantine equation x 2 − kxy + y 2 − 2 n = 0

In this study, we determine when the Diophantine equation x ²?kxy+y ²?2 ⁿ = 0 has an infinite number of positive integer solutions x and y for 0 ? n ? 10. Moreover, we give all positive integer solutions of the same equation for 0 ? n ? 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x ² ? kxy + y ² ? 2 ⁿ = 0.     

On the self matching properties of [jτ]

The graph of [jτ] against j displays self matching in that if we displace this graph by a distance of F_i, then it is found that the displaced graph matches the original graph except at certain isolated points represented by an interesting Fibonacci function. From this it is shown that the frequency of mismatches is the unexpectedly simple expression 1/(τⁱ). The results are proved using lemmas, based on Zeckendorf sums, which have an appeal of their own. These also give simplified solutions to the recurrence of Downey and Griswold. Similar results apply with the Golden Sequence whose jth term is [(j+1)τ]−[jτ].     

On the primality of 2h·3n+1

We consider the primality test of Williams and Zarnke for rational integers of the form 2h·3ⁿ+1. We give an algebraic proof of the test, and we resolve a sign ambiguity. We also show that the conditions of the original test can be relaxed, especially if h is divisible by a power of 2.     

On the number of invariant polynomials of the matrix XAX−1+B

Let F be a field. If A is an n×n matrix over F, we denote by i(A) the number of invariant polynomials of A different from 1. We shall prove that if A,B are n×n matrices over F and t ∈ {1,…,n}, then i(A)+i(B)⩽n+t if and only if there exists a nonsingular matrix X over F such that i(XAX⁻¹+B)⩽t, except in a few cases.     

On the chromatic uniqueness of the graphW(n,n − 2, k)

This paper shows that the graphW(n, n – 2, k) is chromatically unique for any even integern 6 and any integerk 1.     

On the positive integral solutions of the Diophantine equation x 3+by+1−xyz=0

In this paper, we give a formula for the number of positive integral solutions (x,y,z) of the equation x ³+by+1?xyz=0 and also we prove a stronger form of a conjecture of S.P. Mohanty and A.M.S. Ramasamy concerning the number of positive integral solutions (x,y,z) of the equation.     

On the regularity of the growth of the modulus and argument of an entire function in the metric of L p [0, 2π]

Under sufficiently general assumptions, we describe sets of entire functions f, sets of growing functions λ, and sets of complex-valued functions H from L ^p [0, 2π], p ∈ [1, + ∞], for which

Characterization of G 2(q), where 2 < q ≡ −1(mod 3), by order components

We prove that the simple group G ₂(q), where 2 < q ≡ ?1(mod 3), is recognizable by the set of its order components. In other words, we prove that if G is a finite group with OC(G) = OC(G ₂(q)), then G ≌ G ₂(q).