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## A sharp strong maximum principle and a sharp unique continuation theorem for singular minimal hypersurfaces

### Calculus of Variations and Partial Differential Equations (2014-11-01) 51: 799-812 , November 01, 2014

We prove the two theorems of the title, settling two long standing questions in the local theory of singular minimal hypersurfaces. The sharpness of either result is with respect to its hypothesis on the size of the allowable singular sets. The proofs of both theorems rely heavily on the author’s recent regularity and compactness theory for stable minimal hypersurfaces, and on earlier work of Ilmanen, Simon and Solomon–White.

## Some fixed/coincidence point theorems under ( ψ , φ ) -contractivity conditions without an underlying metric structure

### Fixed Point Theory and Applications (2014-10-22) 2014: 1-24 , October 22, 2014

In this paper we prove a coincidence point result in a space which does not have to satisfy any of the classical axioms that define a metric space. Furthermore, the ambient space need not be ordered and does not have to be complete. Then, this result may be applied in a wide range of different settings (metric spaces, quasi-metric spaces, pseudo-metric spaces, semi-metric spaces, pseudo-quasi-metric spaces, partial metric spaces, *G*-spaces, *etc.*). Finally, we illustrate how this result clarifies and improves some well-known, recent results on this topic.

## Stabilization of company’s income modeled by a system of discrete stochastic equations

### Advances in Difference Equations (2014-11-17) 2014: 1-8 , November 17, 2014

The paper deals with a system of difference equations where the coefficients depend on Markov chains. The functional equations for a particular density and the moment equations for the system are derived and used in the investigation of mode stability of company’s income. An application of the results is illustrated by two models.

## Alicia Boole Stott’s models of sections of polytopes

### Lettera Matematica (2014-09-01) 2: 149-154 , September 01, 2014

Alicia Boole Stott (1860–1940) was an amateur mathematician who worked on four-dimensional geometry. She is remembered for finding all three dimensional sections of the four- dimensional polytopes (that is, the analogues of the three-dimensional Platonic solids), and for the discovery of many of the semi-regular polytopes. In this text we give a short biography of her and explain her method to calculate the three-dimensional sections of the four-dimensional polytopes. We illustrate her results by showing pictures of her original models and drawings.

## An Initial-Boundary Value Problem for the Korteweg–de Vries Equation with Dominant Surface Tension

### Acta Applicandae Mathematicae (2014-02-01) 129: 41-59 , February 01, 2014

We consider the initial-boundary value problem (IBVP) for the Korteweg–de Vries equation with zero boundary conditions at *x*=0 and arbitrary smooth decreasing initial data. We prove that the solution of this IBVP can be found by solving two linear inverse scattering problems (SPs) on two different spectral planes. The first SP is associated with the KdV equation. The second SP is self-conjugate and its scattering function is found in terms of entries of the scattering matrix *s*(*k*) for the first SP. Knowing the scattering function, we solve the second inverse SP for finding the potential self-conjugate matrix. Consequently, the unknown object entering coefficients in the system of evolution equations for *s*(*k*,*t*) is found. Then, the time-dependent scattering matrix *s*(*k*,*t*) is expressed in terms of *s*(*k*)=*s*(*k*,0) and of solutions of the self-conjugate SP. Knowing *s*(*k*,*t*), we find the solution of the IBVP in terms of the solution of the Gelfand–Levitan–Marchenko equation in the first inverse SP.

## Hölder Estimates for Nonlocal-Diffusion Equations with Drifts

### Communications in Mathematics and Statistics (2014-12-01) 2: 331-348 , December 01, 2014

We study a class of nonlocal-diffusion equations with drifts, and derive a priori $$\Phi $$ -Hölder estimate for the solutions by using a purely probabilistic argument, where $$\Phi $$ is an intrinsic scaling function for the equation.

## On the Space of Conics on Complete Intersections

### Communications in Mathematics and Statistics (2014-03-01) 2: 33-45 , March 01, 2014

We get sharp degree bound for generic smoothness and connectedness of the space of lines and conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on hypersurfaces. As a consequence, we prove that for a Fano complete intersection $$X$$ with index $$\ge 2$$ , the $$1$$ -Griffiths group generated by algebraic $$1$$ -cycles homologous to $$0$$ modulo algebraic equivalence is trivial, which is a conjecture for general rationally connected varieties.

## Linear Methods in the Proof of Jackson Type Inequalities and Applications to Estimates of Functionals with Two Known Moments

### Journal of Mathematical Sciences (2014-01-01) 196: 498-514 , January 01, 2014

We propose a new method for estimating functionals in terms of higher order moduli of continuity with explicit constants. Using this method, we estimate deviations of linear methods of approximation by entire functions of finite degree and, in particular, by trigonometric polynomials. For illustration of the results, we derive estimates for the Riesz and Akhiezer–Krein–Favard averages. Bibliography: 14 titles.

## Parafermionic Algebras, Their Modules and Cohomologies

### Lie Theory and Its Applications in Physics (2014-01-01) 111: 515-526 , January 01, 2014

We explore the Fock spaces of the parafermionic algebra introduced by H.S. Green. Each parafermionic Fock space allows for a free minimal resolution by graded modules of the graded two-step nilpotent subalgebra of the parafermionic creation operators. Such a free resolution is constructed with the help of a classical Kostant’s theorem computing Lie algebra cohomologies of the nilpotent subalgebra with values in the parafermionic Fock space. The Euler-Poincaré characteristic of the parafermionic Fock space free resolution yields some interesting identities between Schur polynomials. Finally we briefly comment on parabosonic and general parastatistics Fock spaces.

## Serial Group Rings of Finite Groups. p-nilpotency

### Journal of Mathematical Sciences (2014-10-01) 202: 422-433 , October 01, 2014

It is proved that if F is an arbitrary field of characteristic p and G is a finite p-nilpotent group with a cyclic p-Sylow subgroup, then the group ring FG is serial. As a corollary, it is shown that for an arbitrary field F of characteristic 2 and any finite group G, the ring FG is serial if and only if the 2-Sylow subgroup of G is cyclic.