Problems

After a big number of experiments the author conjectured that for all finite Abelian groups G,all homomorphisms f:G^3→ G and all periodic initial conditions g,h:N→ G , the resulting recurrent double sequence can also be generated by an…

Let t_k,q(n) be defined by (5.1) t_k,q(j) = q^j, for 0 ≤ j ≤ k-1 t_k,q(n) = q ∑_l=0^k-2 (q-1)^l t_k,q(n-(l+2)), for n ≥ k. Then, S_F_q(T_2,3,…,k(n)) = t_k,q(n) for all values of n ≥ k.

Find other possible values of the parameter d and the corresponding d-antimagic labeling of type (1, 1, 1) for the generalized Petersen graph P(n, 2).

Construct and analyze analogues of the Hanoi graph H_n in variants 4 and 5. Is there a generalization of the Lucas Correspondence that works here?

Can you show F(n; 2)< (1+ε)n/2 ? What about larger values of t ?

For every 2-coloring of the positive integers there exist positive integers a,d such that the elements of the arithmetic progression a,a+d,...,a+d^2 all are colored the same.

Let C be a projective [n,k]_q three-weight code with non-zero weights w_1 < w_2 < w_3 satisfying w_1 + w_2 + w_3 = 3(1-1/q)n. Then w_2 = (1-1/q)n. Moreover, w_1 = w_2 - t and w_3 = w_2 + t, where t is a power of the characteristic p of F_q.

For any ℓ ∈N and any (not necessarily finite) partition N=∪_i=1^∞C_i , there exists d ∈N such that either (i) for some i_0∈N we have C_i_0∩(C_i_0-d)∩(C_i_0-2d)∩… ∩(C_i_0-ℓ d)≠∅ , or (ii) for every i ∈N we have (C_i-jd)∩(C_i-kd)=∅ for all…

If k is a power of a prime, P_L_k(G,k)=P_DP(G,k).

Conjecture 1.6.1. The polynomial f_m(b,q) has the form f_m(b,q)=∑_i=0^binomm2(1-q)^m-y^(i)g_m,i(q)b^i where y(n)=⌊frac√8n+12⌋ and g_m,i(q) are polynomials. Further, with <_k^n> denot ing the Eulerian numbers ^3…

B(n, m) = n(m-1) + 1.

Based upon the results generated from our Sage script, we submit as a conjecture that these graphs constructed be the smallest graphs (by order) that have characteristic-dependent well-covered dimension for any given characteristic.

Let δ≥ 3 be an integer. Does there exist a δ -chromatic quadruple system Q such that χ(K(Q))=δ ?

the existence of such an f has been proved, but uniqueness in T_0 has not.

Let (X,B,μ) be a σ -finite measure space and T:X → X a measure preserving transformation. If A ∈B , then there exists n ∈N with μ(A ∩ T^-nA ∩ T^-2nA ∩… ∩ T^-ℓ nA)>0,or (5) μ(T^-inA ∩ T^-jnA)=0 ∀0 ≤ i<j ≤ℓ. (6)

This results lead us to establish a conjecture that gives us an algebraic and a combinatorial description of the sandpile group of the cone of the hypercube Q_d of dimension d. More precisely, K(c(Q_d)) ≅ bigoplus_i=1^d Z_2i+1^binomdi =…

For integers n, r and a prime p satisfying r < p, we have ex(n, K_r, C_≥ p^prime) ≤ n-1/p-2 binomp-1r. Equality holds only for connected n-vertex graphs consisting of n-1/p-2 maximal 2-connected blocks each isomorphic to K_p-1.

For n≥2 B(K_2,n,Z_2n)≤4n-3 .

Note that the divisibility conditions in (22) should be equivalent to those in (23) if a t-(n,k,λ) exists. It is open if they are equivalent.

We further conjecture that an elliptic quadric is incident with m modulo q points of an m-ovoid of Q(4, q).