The formula of the solution for some classes of initial boundary value problems for the hyperbolic equation with two independent variables

A new representation is proved of the solutions of initial boundary value problems for the equation of the form u _xx (x, t) + r(x)u _x (x, t) ? q(x)u(x, t) = u _tt (x, t) + μ(x)u _t (x, t) in the section (under boundary conditions of the 1st, 2nd, or 3rd type in any combination). This representation has the form of the Riemann integral dependent on the x and t over the given section.     

Some remarks on a conjecture in parameter adaptive control

A.S. Morse has raised the following question: Do there exist differentiable functions

$\text{f:R}{}^2\text{→ R}\text{and}\text{g:R}{}^2\text{→ R}$

with the property that for every nonzero real number λ and every (x₀, y₀) ∈ $\text{R}$² the solution (x(t),y(t)) of

$\text{x}\text{?}\text{(t) = x(t) + λf(x(t),y(t))}$

,

$\text{y}\text{?}\text{(t) = g(x(t),y(t))}$

,

$\text{x(0) = x}{}_0\text{, y(0) = y}{}_0$

, is defined for all t ? 0 and satisfies

$\text{lim}\text{t → + ∞}$

and y(t) is bounded on [0,∞)? We prove that the answer is yes, and we give explicit real analytic functions f and g which work. However, we prove that if f and g are restricted to be rational functions, the answer is no.     

On the lower bounds of the second order nonlinearities of some Boolean functions

The rth order nonlinearity of a Boolean function is an important cryptographic criterion in analyzing the security of stream as well as block ciphers. It is also important in coding theory as it is related to the covering radius of the Reed-Muller code R(r,n). In this paper we deduce the lower bounds of the second order nonlinearities of the following two types of Boolean functions:

1.

with d=2^2r+2^r+1 and , where n=6r.

2.

, where x,y∈F_2^t,n=2t,n?6 and i is an integer such that 1?i<t,gcd(2^t-1,2ⁱ+1)=1.

For some λ, the functions of the first type are bent functions, whereas Boolean functions of the second type are all bent functions, i.e., they possess the maximum first order nonlinearity. It is demonstrated that in some cases our bounds are better than the previously obtained bounds.     

Asymptotic and oscillatory behavior of solutions of general nonlinear difference equations of second order

The asymptotic and oscillatory behavior of solutions of some general second-order nonlinear difference equations of the form

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=0 nZ,

is studied. Oscillation criteria for their solutions, when “p_n” is of constant sign, are established. Results are also presented for the damped-forced equation

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=e_n nZ.

Examples are inserted in the text for illustrative purposes.     

Clique Numbers of Random Subgraphs of Some Distance Graphs

We consider a class of graphs G(n, r, s) = (V (n, r),E(n, r, s)) defined as follows:

$$V(n,r) = \{ x = ({x_{1,}},{x_2}...{x_n}):{x_i} \in \{ 0,1\} ,{x_{1,}} + {x_2} + ... + {x_n} = r\} ,E(n,r,s) = \{ \{ x,y\} :(x,y) = s\} $$

where (x, y) is the Euclidean scalar product. We study random subgraphs G(G(n, r, s), p) with edges independently chosen from the set E(n, r, s) with probability p each. We find nontrivial lower and upper bounds on the clique number of such graphs.

     

A further result on the oscillation of delay difference equations

In this paper, we are concerned with the delay difference equations of the form

(*)

y_n+1 − y_n + p_ny_n−k = 0, N = 0, 1, 2, …,

(*)where p_n ≥ 0 and k is a positive integer. We prove by using a new technique that

guarantees that all solutions of equation (*) oscillate, which improves many previous well-known results. In particular, our theorems also fit the case where Σⁿ⁻¹_i=n−kp_i ≤ k^k+1/(k + 1)^k+1. In addition, we present a nonoscillation sufficient condition for equation (*).     

Oscillation criteria of certain nonlinear partial difference equations

This paper is concerned with the nonlinear partial difference equation with continuous variables

,where a, σ_i, τ_i are positive numbers, h_i(x, y, u) ε C(R⁺ × R⁺ × R, R), uh_i(x, y, u) > 0 for u ≠ 0, h_i is nondecreasing in u, i = 1, …, m. Some oscillation criteria of this equation are obtained.     

Fuzzy approximately cubic mappings

We establish some stability results concerning the cubic functional equation

f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)     

Parabolic equations with double variable nonlinearities

The paper is devoted to the study of the homogeneous Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions:

ut=div(a(x,t,u)|u^|α(x,t)|∇u^|p(x,t)−2∇u)+f(x,t)     

Optimal Path Embedding in the Exchanged Crossed Cube

The (s + t + 1)-dimensional exchanged crossed cube, denoted as ECQ(s, t), combines the strong points of the exchanged hypercube and the crossed cube. It has been proven that ECQ(s, t) has more attractive properties than other variations of the fundamental hypercube in terms of fewer edges, lower cost factor and smaller diameter. In this paper, we study the embedding of paths of distinct lengths between any two different vertices in ECQ(s, t). We prove the result in ECQ(s, t): if s ≥ 3, t ≥ 3, for any two different vertices, all paths whose lengths are between \( \max \left\{9,\left\lceil \frac{s+1}{2}\right\rceil +\left\lceil \frac{t+1}{2}\right\rceil +4\right\} \) and 2 ^s+t+1 ? 1 can be embedded between the two vertices with dilation 1. Note that the diameter of ECQ(s, t) is \( \left\lceil \frac{s+1}{2}\right\rceil +\left\lceil \frac{t+1}{2}\right\rceil +2 \). The obtained result is optimal in the sense that the dilations of path embeddings are all 1. The result reveals the fact that ECQ(s, t) preserves the path embedding capability to a large extent, while it only has about one half edges of CQ _n .     

Sleeping on the Job: Energy-Efficient and Robust Broadcast for Radio Networks

We address the problem of minimizing power consumption when broadcasting a message from one node to all the other nodes in a radio network. To enable power savings for such a problem, we introduce a compelling new data streaming problem which we call the Bad Santa problem. Our results on this problem apply for any situation where: (1) a node can listen to a set of n nodes, out of which at least half are non-faulty and know the correct message; and (2) each of these n nodes sends according to some predetermined schedule which assigns each of them its own unique time slot. In this situation, we show that in order to receive the correct message with probability 1, it is necessary and sufficient for the listening node to listen to a \(\Theta(\sqrt{n})\) expected number of time slots. Moreover, if we allow for repetitions of transmissions so that each sending node sends the message O(log?^? n) times (i.e. in O(log?^? n) rounds each consisting of the n time slots), then listening to O(log?^? n) expected number of time slots suffices. We show that this is near optimal.We describe an application of our result to the popular grid model for a radio network. Each node in the network is located on a point in a two dimensional grid, and whenever a node sends a message m, all awake nodes within L _∞ distance r receive m. In this model, up to \(t<\frac{r}{2}(2r+1)\) nodes within any 2r+1 by 2r+1 square in the grid can suffer Byzantine faults. Moreover, we assume that the nodes that suffer Byzantine faults are chosen and controlled by an adversary that knows everything except for the random bits of each non-faulty node. This type of adversary models worst-case behavior due to malicious attacks on the network; mobile nodes moving around in the network; or static nodes losing power or ceasing to function. Let n=r(2r+1). We show how to solve the broadcast problem in this model with each node sending and receiving an expected \(O(n\log^{2}{|m|}+\sqrt{n}|m|)\) bits where |m| is the number of bits in m, and, after broadcasting a fingerprint of m, each node is awake only an expected \(O(\sqrt{n})\) time slots. Moreover, for t≤(1?ε)(r/2)(2r+1), for any constant ε>0, we can achieve an even better energy savings. In particular, if we allow each node to send O(log?^? n) times, we achieve reliable broadcast with each node sending O(nlog?²|m|+(log?^? n)|m|) bits and receiving an expected O(nlog?²|m|+(log?^? n)|m|) bits and, after broadcasting a fingerprint of m, each node is awake for only an expected O(log?^? n) time slots. Our results compare favorably with previous protocols that required each node to send Θ(|m|) bits, receive Θ(n|m|) bits and be awake for Θ(n) time slots.     

On the Smallest Size of an Almost Complete Subset of a Conic in PG(2, <Emphasis Type="Italic">q</Emphasis>) and Extendability of Reed–Solomon Codes

Abstract—In the projective plane PG(2, q), a subset S of a conic C is said to be almost complete if it can be extended to a larger arc in PG(2, q) only by the points of C \ S and by the nucleus of C when q is even. We obtain new upper bounds on the smallest size t(q) of an almost complete subset of a conic, in particular,

$$t(q) < \sqrt {q(3lnq + lnlnq + ln3)} + \sqrt {\frac{q}{{3\ln q}}} + 4 \sim \sqrt {3q\ln q} ,t(q) < 1.835\sqrt {q\ln q.} $$

The new bounds are used to extend the set of pairs (N, q) for which it is proved that every normal rational curve in the projective space PG(N, q) is a complete (q+1)-arc, or equivalently, that no [q+1,N+1, q?N+1]_q generalized doubly-extended Reed–Solomon code can be extended to a [q + 2,N + 1, q ? N + 2]_q maximum distance separable code.

     

On weak isometries of Preparata codes

Two codes C ₁ and C ₂ are said to be weakly isometric if there exists a mapping J: C ₁ → C ₂ such that for all x, y in C ₁ the equality d(x, y) = d holds if and only if d(J(x), J(y)) = d, where d is the code distance of C ₁. We prove that Preparata codes of length n ≥ 2¹² are weakly isometric if and only if the codes are equivalent. A similar result is proved for punctured Preparata codes of length at least 2¹⁰ ? 1.     

The complexity of changing colourings with bounded maximum degree

Let G be a graph, x,y∈V(G), and ?:V(G)→[k] a k-colouring of G such that ?(x)=?(y). If then the following question is NP-complete: Does there exist a k-colouring ?^′ of G such that ?^′(x)≠?^′(y)? Conversely, if then the problem is polynomial time.     

Polylogarithms and the Asymptotic Formula for the Moments of Lebesgue’s Singular Function

Recall that Lebesgue’s singular function L(t) is defined as the unique solution to the equation L(t) = qL(2t) + pL(2t ? 1), where p, q > 0, q = 1 ? p, p ≠ q. The variables M _n = ∫₀¹t ⁿ dL(t), n = 0,1,… are called the moments of the function The principal result of this work is \({M_n} = {n^{{{\log }_2}p}}{e^{ - \tau (n)}}(1 + O({n^{ - 0.99}}))\), where the function τ(x) is periodic in log₂x with the period 1 and is given as \(\tau (x) = \frac{1}{2}1np + \Gamma '(1)lo{g_2}p + \frac{1}{{1n2}}\frac{\partial }{{\partial z}}L{i_z}( - \frac{q}{p}){|_{z = 1}} + \frac{1}{{1n2}}\sum\nolimits_{k \ne 0} {\Gamma ({z_k})L{i_{{z_k} + 1}}( - \frac{q}{p})} {x^{ - {z_k}}}\), \({z_k} = \frac{{2\pi ik}}{{1n2}}\), k ≠ 0. The proof is based on poissonization and the Mellin transform.     

Generalized Tweakable Even-Mansour Cipher and Its Applications

This paper describes a generalized tweakable blockcipher HPH (Hash-Permutation-Hash), which is based on a public random permutation P and a family of almost-XOR-universal hash functions \( \mathcal{H}={\left\{ HK\right\}}_{K\in \mathcal{K}} \) as a tweak and key schedule, and defined as y = HPH_K((t₁, t₂), x) = P(x ⊕ H_K(t₁)) ⊕ H_K(t₂), where K is a key randomly chosen from a key space \( \mathcal{K} \), (t₁, t₂) is a tweak chosen from a valid tweak space \( \mathcal{T} \), x is a plaintext, and y is a ciphertext. We prove that HPH is a secure strong tweakable pseudorandom permutation (STPRP) by using H-coefficients technique. Then we focus on the security of HPH against multi-key and related-key attacks. We prove that HPH achieves both multi-key STPRP security and related-key STPRP security. HPH can be extended to wide applications. It can be directly applied to authentication and authenticated encryption modes. We apply HPH to PMAC1 and OPP, provide an improved authentication mode HPMAC and a new authenticated encryption mode OPH, and prove that the two modes achieve single-key security, multi-key security, and related-key security.     

Oscillation theorems for second-order nonlinear neutral differential equations

The purpose of this paper is to study the oscillation of the second-order neutral differential equations of the form

(a(t)[z^′(t)]^γ)^′+q(t)x^β(σ(t))=0,     

Oscillation criteria for second order quasilinear neutral delay differential equations on time scales

In this paper, we consider the second-order quasilinear neutral dynamic equation

(r(t)|Z^Δ(t)|^α−1Z^Δ(t))^Δ+q(t)|x(δ(t))|^β−1x(δ(t))=0,     

Factorization of disconjugate higher-order Sturm-Liouville difference operators

Using a recently proved equivalence between disconjugacy of the 2n^th-order difference equation

and solvability of the corresponding Riccati matrix difference equation, it is shown that the equation L(y) = 0 is disconjugate on a given interval if and only if the operator L admits the factorization of the form

L(y)_k+n=M^*(c_kM(y)_k)_k+n,

where M and its adjoint M* are certain n^th-order difference operators and c_k is a sequence of positive numbers.     

The asymptotic self-similar behavior for the quasilinear heat equation with nonlinear boundary condition

In this paper the quasilinear heat equation with the nonlinear boundary condition is studied. The blow-up rate and existence of a self-similar solution are obtained. It is proved that the rescaled function

v(y,t)=(T−t)^{1/(2p+α−2)}u((T−t)^{(p−1)/(2p+α−2)}y,t),