SUMMARYIn this paper, we investigate a relationship between the length-decreasing self-reducibility and the many-one-like reducibilities for partial multivalued functions. We show that if any parsimonious (manyone or metric many-one) complete function for NPMV (or NPMV g ) is length-decreasing self-reducible, then any function in NPMV (or NPMV g ) has a polynomial-time computable refinement. This result implies that there exists an NPMV (or NPMV g )-complete function which is not lengthdecreasing self-reducible unless P = NP.
Cascade hydroxyl radical generating hydrogel reactor structures including a chemotherapeutic agent are invented for multiple treatment of breast cancer. Glucose oxidase (GOx) and cupric sulfate (Cu) are introduced for transforming accumulated glucose (in cancer cells) to hydroxyl radicals for starvation/chemodynamic therapy. Cu may also suppress cancer cell growth via cuproptosis‐mediated cell death. Berberine hydrochloride (BER) is engaged as a chemotherapeutic agent in the hydrogel reactor for combining with starvation/chemodynamic/cuproptosis therapeutic modalities. Moreover, Cu is participated as a gel crosslinker by coordinating with catechol groups in hyaluronic acid‐dopamine (HD) polymer. Controlling viscoelasticity of hydrogel reactor can extend the retention time following local injection and provide sustained drug release patterns. Low biodegradation rate of designed HD/BER/GOx/Cu hydrogel can reduce dosing frequency in local cancer therapy and avoid invasiveness‐related inconveniences. Especially, it is anticipated that HD/BER/GOx/Cu hydrogel system can be applied for reducing size of breast cancer prior to surgery as well as tumor growth suppression in clinical application.
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