We investigate singularities of height functions on generalized cuspidal edges. In particular, we study singularities of height functions in the normal and conormal directions. Moreover, we examine the bifurcation set of the family of height functions.
A Lehmer number modulo an odd prime number p is a residue class a ∈𝔽_p^× whose multiplicative inverse a̅ is of opposite parity with a. Lehmer numbers that are also quadratic residues (respectively, non-residues, primitive roots and non-primitive roots) are called Lehmer quadratic residues (respectively, Lehmer non-residues, Lehmer primitive roots and Lehmer non-primitive roots). In this article, we prove that for any prime p > e^27 (respectively, for any prime p satisfying loglog p > log 2.62/(1/2 - ϵ ) with ϵ∈ [1/5, 1/2) and ϕ (p-1)/ (p-1) ≤ 1/2 - ϵ ), there exists Lehmer quadratic residue and Lehmer non-residue (respectively, Lehmer non-residue which are non-primitive root) modulo p. We also provide explicit estimates for the difference between the number of such numbers and their “expected number”. We also study the Lehmer number problem by varying primes p. In particular, for a fixed integer a ≥ 2 , we prove the existence of infinitely many prime numbers p for which a is a Lehmer number modulo p. In particular, we obtain a bound for the relative density of the set of all primes for which a is a Lehmer number, in terms of ϕ (a) . Moreover, when a = 2 or 3, we explicitly find congruence classes for such primes.
Let ℋ and 𝒦 be complex separable Hilbert spaces, and consider T= [ aI_ℋ A; cA^* bI_𝒦 ], A∈ℬ(𝒦,ℋ), a,b,c∈ℂ. Using the polar decomposition of A, we separate the coupled part of T from the scalar components supported on the defect spaces A^* and A . The spectral theorem then reduces the coupled part to the family M_t= [ a t; ct b ], t∈σ( . |A|| _( A)^⊥) . This yields exact formulas for both the minimum modulus and the essential minimum modulus of T. We characterize minimum attainment in terms of the spectral measure of the reduced modulus and obtain necessary and sufficient conditions for boundedness below and upper semi-Fredholmness. We further show that a strict gap between the two moduli forces the ordinary minimum to be an isolated eigenvalue of |T| of finite multiplicity. Applications are given to triangular operators, partial isometries, compact off-diagonal operators, and Buckholtz-type operators.
The study introduces an idea that addresses the fractional bilevel optimization problem, which involves fractional objectives at both the upper and lower levels. The upper-level fractional problem is tackled using a standard parametric reformulation technique commonly adopted in the existing literature. The lower-level fractional problem adopts the same parametric reformulation approach. The key idea is to represent the fractional lower-level problem as a simple optimization problem, where the optimal solution set equals the actual set. This approach helps us to derive necessary and sufficient optimality conditions along with the concept of convexificators and generalized convexity. This formulation provides a structured and computationally viable framework to solve such complex hierarchical optimization problems more efficiently. Numerical examples illustrate the theoretical results.
A weighted sparse Kaczmarz method with adaptive momentum is developed for the sparse solution of the linear system of equations, where the momentum parameter is dynamically determined by minimizing the Bregman distance. The linear convergence of the proposed method is provided, the advantage of the momentum mechanism is supported by theoretical analysis. Numerical experiments verify the efficiency of the proposed method in terms of the number of iterations and CPU time.
